Abstract
Computational Fluid Dynamics (CFD) is an important design tool in engineering and also a substantial research tool in various physical sciences as well as in biology. The objective of this book is to provide university students with a solid foundation for understanding the numerical methods employed in todays CFD and to familiarise them with modern CFD codes by hands-on experience. It is also intended for engineers and scientists starting to work in the field of CFD or for those who apply CFD codes. Due to the detailed index, the text can serve as a reference handbook too. Each chapter includes an extensive bibliography, which provides an excellent basis for further studies. The accompanying CD-ROM contains the sources of 1-D and 2-D Euler and Navier-Stokes flow solvers (structured and unstructured) as well as of grid generators. Provided are also tools for Von Neumann stability analysis of 1-D model equations. Finally, the CD-ROM includes the source code of a dedicated visualisation software with graphical user interface.
... The full set of (Favre-and) Reynolds-averaged Navier-Stokes equations, solved in the framework of computational fluid dynamics, reads [11]: ...
... Employing the eddy-viscosity hypothesis introduced by Boussinesq, the components of the Reynolds-stress tensor can be expressed as [11]: ...
... For the numerical flow simulations presented within this thesis, the k − ω Shear Stress Transport (SST) turbulence model derived by Menter [77] is used. The model is based on two additional transport equations for the turbulent kinetic energy k and the specific turbulent dissipation rate ω [11]: ...
Hybrid laminar flow control (HLFC) and variable camber (VC) are promising techniques for reducing aerodynamic drag of transport aircraft. This thesis numerically investigates the synergy potential between HLFC and VC. Using linear stability theory and transition turbulence models, the synergistic potential on different integration levels throughout multiple parts of the flight envelope is confirmed. The results are also used to formulate reduced order models, enabling multi-fidelity analyses on overall aircraft level.
... It utilizes a cell-centered finite volume method for solving a set of partial differential equations (PDEs) with standard Gaussian finite volume integration. Non-bounded second-order numerical schemes (Blazek 2015) are used for discretizing the convective and diffusion terms. So the Gauss linear upwind scheme is applied for the velocity divergence and kinetic energy transfer terms, while the Gaussian VanLeer scheme is used for the volume fraction equation. ...
... Laplacian terms are evaluated using the Gauss linear corrected scheme. The backward implicit second-order method (Blazek 2015) discretizes the time derivative terms. To ensure stability, the solution process is iterated multiple times at each time step, with a maximum Courant number of 0.3 and a minimum mean time step of 0.2 microseconds. ...
... To ensure stability, the solution process is iterated multiple times at each time step, with a maximum Courant number of 0.3 and a minimum mean time step of 0.2 microseconds. The pressure-velocity coupling is achieved using the Pressure-Implicit with Splitting of Operators (PISO) algorithm (Blazek 2015), and the calculations are performed iteratively within an inner loop. ...
Pressure-swirl atomizers (PSAs) are widely used in industry due to their advantages, necessitating a comprehensive analysis of internal and external flow dynamics. This study employs the interIsoFoam solver in OpenFOAM with the k-Eqn LES model to investigate transient three-dimensional flow in a PSA and the primary atomization of the exiting conical liquid sheet. Validation against experimental data demonstrates good agreement, with average errors of 0.05% (spray cone angle), 3.4% (discharge coefficient), and 0.07% (air core diameter). The simulation explores the complex internal flow phenomena of the atomizer and the outlet liquid sheet, including the air core structure and mechanisms of liquid sheet breakup. FFT spectral analysis identifies the dominant high-energy frequencies and reveals similarities between the frequencies of the internal flow within the atomizer and those of the exiting liquid sheet. The Proper Orthogonal Decomposition (POD) technique captures the energetic structures of the atomizer’s internal and external flows, indicating that, similar to the temporal domain, the energetic phenomena of internal flow impact atomization in the frequency domain. The POD analysis underscores the significance of energetic internal flow features, such as helical air PVC, strong shear layers, and liquid vortices, in shaping external flow structures like strong shear layers, surface waves, inner recirculation zones, and air vortices. This culminates in the formation of a stagnation point and the liquid sheet breakup at a height of . The study emphasizes the influence of internal flow structures on atomization mechanisms and liquid sheet breakup, underscoring the pivotal role of internal energetic flow features in the process.
... The curvature of the contraction walls, whether concave or convex, affects the streamline curvature near the walls, which in turn influences boundary layer separation. The impact of these effects on turbulence is captured using a curvature correction model, 40,42 although the boundary layer thickness remains relatively small in the present study. Another physical effect not captured by eddyviscosity-based models is the secondary flow induced by turbulence, which arises due to the anisotropy of normal stresses, particularly near wall corners. ...
... Another physical effect not captured by eddyviscosity-based models is the secondary flow induced by turbulence, which arises due to the anisotropy of normal stresses, particularly near wall corners. 40,42 For instance, in turbulent flow within square or rectangular ducts, four pairs of large-scale vortical structures, known as secondary flows, typically form. 43,44 These secondary flows, manifesting as nonzero mean velocities within the transverse planes adjacent to each corner, can significantly influence the overall flow field within a contraction. ...
... Excluding them can lead to inaccuracies in CFD predictions, particularly in the flow topology. To address these issues, both curvature correction and corner correction 40,42 formulations are incorporated into the simulations to improve accuracy. Without these corrections, the flow near the corners may undergo premature separation due to the adverse pressure gradient. ...
This study presents a systematic parametric investigation aimed at optimizing the contraction design for a large low-speed wind tunnel. The numerical study is undertaken by relating the geometric and performance parameters of a contraction nozzle to satisfy a certain set of design criteria. The Computational Fluid Dynamics (CFD) simulations were performed in ANSYS Fluent by solving the Reynolds Averaged Navier-Stokes (RANS) approach coupled with the Shear Stress Transport (SST) k-ω turbulence model. Contraction design charts were developed for three different test section flow speeds (10 m/s, 25 m/s, and 75 m/s). The range of geometrical parameters explored includes contraction length (L/D) from 0.8 to 1.6, match point location (X) from 0.2 to 0.8, and polynomial power factors ( n 1 and n 2 ) ranging from 3 to 9. An optimal combination of these parameters effectively prevents boundary layer flow separation around the entrance, while also minimizing boundary layer thickness and velocity field nonuniformity downstream of the exit. Notably, transitioning from a square-to-rectangular cross-section to an octagon-to-octagon cross-section significantly enhances the flow quality in the test section. The contraction with an octagonal cross-section, characterized by n 1 = 3, n 2 = 6, L/D = 1.02, and X = 0.575, meets all design criteria and is proposed as the optimal choice for a large low-speed wind tunnel with a contraction ratio (CR) of 9.
... where CFL is the Courant-Friedrichs-Lewy number and l minedge,P is the length of the shortest edge connected with the examined node P. This acceleration technique allows for the maximum permissible time step to be employed for each node and consequently to improve the convergence rate of the relaxation procedure (Blazek, 2001). Finally, the corrections to F are computed iteratively implementing (at each external time step) the explicit second-order four-stage Runge-Kutta method (RK(4)) (Lallemand, 1988), hence gradually approximating the final steady state. ...
... Finally, the corrections to F are computed iteratively implementing (at each external time step) the explicit second-order four-stage Runge-Kutta method (RK(4)) (Lallemand, 1988), hence gradually approximating the final steady state. The numerical solver is enhanced with an agglomeration multigrid scheme to further accelerate the solution procedure, especially in large-scale problems (Blazek, 2001;Nishikawa et al., 2010;Lygidakis et al., 2016). Although this approach was initially introduced in a three-dimensional CFD solver (Lygidakis et al., 2016), it was incorporated in the current two-dimensional Laplacian algorithm in an almost straightforward manner, mainly due to its edge-based formulation. ...
... solution is obtained with the Full Approximation Scheme (FAS) in a V-cycle process. According to this methodology Equation 11 is solved only at the initial finest mesh, whereas at the coarser ones, approximate formulations of it are relaxed (Blazek, 2001;Nishikawa et al., 2010). The interaction between each two successive spatial levels is achieved via the restriction of F values and flux balances, computed at the centres of control cells, from the finer to the coarser resolution, as well as with the prolongation of the corresponding updated corrections to F from the coarser to the finer one. ...
The automatic deformation of the computational mesh along with the deformed geometry in design optimization cycles is a valuable procedure, as it reduces the required time for the construction of new meshes. The introduction of harmonic coordinates for the deformation of objects included within a closed mesh (cage) has been introduced in computer graphics. Harmonic coordinates result from solutions to the Laplace’s equation (harmonic functions) using a numerical solver. In this work, a modification to the classical harmonic coordinates’ concept is introduced for the deformation of 2D geometries (and the corresponding computational mesh) which are defined as B-spline curves. The B-spline basis functions are used as harmonic functions along the mesh boundary, being also the geometry to be deformed. Thus, any deformation of the B-spline boundary, through the movement of the curve’s control points, can be successfully propagated to the interior of the computational domain, resulting in the concurrent and conformable modification of the B-spline boundary and the entire computational mesh. For the computation of harmonic coordinates a node-centered Finite-Volume based Laplace solver for unstructured meshes is used, enhanced with an agglomeration multigrid scheme. The proposed method is applied and assessed for the shape and mesh morphing of airfoils.
... These deeds of physicality depend on a captivating wonder called the Magnus effect. [1][2][3][4] The Magnus effect portrays the era of a sidelong constrain on a turning question moving through a liquid (fluid or gas). This constrain acts opposite to the course of movement and the pivot of turn, causing the protest to veer off from its straight way. ...
... Accurately modelling the complex interactions between the ball's spin, the formation of vortices, and their influence on the pressure field requires advanced computational fluid dynamics (CFD) techniques. 1,2 For simplified modelling, see Appendix II. ...
... I. Navier-Stokes Equations: These equations govern the motion of viscous fluids. Solving them for a turbulent flow around a spinning football requires advanced computational fluid dynamics (CFD) techniques.1,2 II. ...
The present review article can be considered as an exploration in recreational mathematics. The Magnus effect, a phenomenon arising from the interaction between a spinning object and a fluid, plays a crucial role in various sports. This paper delves specifically into its impact on football (soccer). We explore the fundamental principles behind the Magnus effect, explaining how a spinning ball creates a pressure difference that leads to a lateral force, causing the ball to deviate from its expected trajectory. Additionally, the discussion explores the limitations of the Magnus effect and how external conditions can affect its influence. Finally, the paper concludes by highlighting the importance of mastering spin techniques to achieve tactical advantage and enhance ball control.
... The general equation forms are presented in Equations (1) to (9) [13]. Given that heat generation, buoyancy forces, and gravity effects are disregarded, the right-hand side of Equation (1), denoted as J, exclusively represents the source terms related to turbulence equations [15,46]: ...
... where, U is general (scalar) flow variable, u, v, and w are cartesian velocity components, ρ is density, P is pressure, T is temperature, µ is dynamic viscosity coefficient, τ ij is components of viscous stress tensor, and e is the total specific energy [2,15,46]. The shear-stress transport was used to analyze boundary layer separation, shock generation, and vortex areas in detail. ...
... In the field of natural sciences, physical systems governed by partial differential equations (PDEs) have found widespread applications across disciplines including biology, chemistry, meteorology, etc. [1][2][3][4][5]. Although numerical methods have been regarded as reliable tools for modeling these systems, the use of Direct Numerical Simulation (DNS) faces significant hurdles due to inherent limitations. ...
... Computational fluid dynamics. Computational Fluid Dynamics (CFD) is a branch of fluid mechanics that uses numerical methods and algorithms to analyze and solve problems involving fluid flow [1][2][3][4][5]. When faced with complex problems, unacceptable time and computational costs are the primary obstacles limiting the application of CFD. ...
Simulation of spatiotemporal systems governed by partial differential equations is widely applied in fields such as biology, chemistry, aerospace dynamics, and meteorology. Traditional numerical methods incur high computational costs due to the requirement of small time steps for accurate predictions. While machine learning has reduced these costs, long-term predictions remain challenged by error accumulation, particularly in scenarios with insufficient data or varying time scales, where stability and accuracy are compromised. Existing methods often neglect the effective utilization of multi-scale data, leading to suboptimal robustness in predictions. To address these issues, we propose a novel multi-scale learning framework, namely, the Physics-Informed Multi-Scale Recurrent Learning (PIMRL), to effectively leverage multi-scale data for spatiotemporal dynamics prediction. The PIMRL framework comprises two modules: the micro-scale module embeds physical knowledge into neural networks via pretraining, and the macro-scale module adopts a data-driven approach to learn the temporal evolution of physics in the latent space. Experimental results demonstrate that the PIMRL framework consistently achieves state-of-the-art performance across five benchmark datasets ranging from one to three dimensions, showing average improvements of over 9\% in both RMSE and MAE evaluation metrics, with maximum enhancements reaching up to 80%.
... where the dynamic Smagorinsky coefficient C D for element e is given by [61,62]: ...
... where the operation ⟨·⟩ e is given by Eq. (50). The L ij and M ij tensors are given by [61,62,42]: ...
The performance of the nonlinearly stable flux reconstruction (NSFR) schemes for resolving subsonic viscous turbulent free-shear flows is investigated. The schemes are extensively verified for the direct numerical simulation (DNS) of the Taylor-Green Vortex (TGV) problem. Several under-resolved simulations of the TGV problem are conducted to assess the performance of NSFR for large eddy simulation that is implicitly filtered and fully implicit (ILES). Increasing the flux reconstruction correction parameter ensures that NSFR is stable and accurate for ILES while allowing for larger explicit time-steps. The entropy-stable schemes implemented with sum-factorization for tensor and Hadamard products are shown to be more cost-effective than classical DG with over-integration. The choice of the two-point (TP) numerical flux does not impact the solution and the use of standard eddy-viscosity-based sub-grid scale models does not yield improvements for the problem considered. From the DNS results, the pressure dilatation-based dissipation rate for the nonlinearly stable schemes is consistent with literature when computed from the kinetic energy (KE) budget terms, while spurious oscillations are seen when the term is directly computed. The magnitude of these oscillations is significantly lower for a collocated scheme and are effectively eliminated with the addition of Roe upwind dissipation to the TP numerical flux. Therefore, these oscillations are believed to be associated with the treatment of the face terms in nonlinearly stable schemes. It is shown that oversampling the velocity field is necessary for obtaining accurate turbulent KE (TKE) spectra and eliminates an apparent pile-up of TKE at the smallest resolved scales. Lastly, the TKE spectra for a decaying homogeneous isotropic turbulence case are in good agreement with experiment measurements and computational results in the literature.
... Despite the critical importance of CFD across scientific and engineering domains (Blazek, 2015), a significant gap exists in providing intelligent tools that can automate the complex simulation workflow while maintaining physical and mathematical consistency. The democratization of CFD through intelligent automation (Chen et al., 2024a) has been a long-standing goal in the field. ...
Computational Fluid Dynamics (CFD) is an essential simulation tool in various engineering disciplines, but it often requires substantial domain expertise and manual configuration, creating barriers to entry. We present Foam-Agent, a multi-agent framework that automates complex OpenFOAM-based CFD simulation workflows from natural language inputs. Our innovation includes (1) a hierarchical multi-index retrieval system with specialized indices for different simulation aspects, (2) a dependency-aware file generation system that provides consistency management across configuration files, and (3) an iterative error correction mechanism that diagnoses and resolves simulation failures without human intervention. Through comprehensive evaluation on the dataset of 110 simulation tasks, Foam-Agent achieves an 83.6% success rate with Claude 3.5 Sonnet, significantly outperforming existing frameworks (55.5% for MetaOpenFOAM and 37.3% for OpenFOAM-GPT). Ablation studies demonstrate the critical contribution of each system component, with the specialized error correction mechanism providing a 36.4% performance improvement. Foam-Agent substantially lowers the CFD expertise threshold while maintaining modeling accuracy, demonstrating the potential of specialized multi-agent systems to democratize access to complex scientific simulation tools. The code is public at https://github.com/csml-rpi/Foam-Agent
... These simulations enable the analysis of flow dynamics within the nozzle and the optimization of its geometry to achieve desired output flow parameters. In the present study, the objective is to minimize the axial flow pressure to reduce electron beam scattering [17][18][19][20]. ...
The ongoing research in Environmental Scanning Electron Microscopy (ESEM) is contributed to in this paper. Specifically, this study investigates supersonic flow in a nozzle aperture under low-pressure conditions at the continuum mechanics boundary. This phenomenon is prevalent in the differentially pumped chamber of an ESEM, which separates two regions with a significant pressure gradient using an aperture with a pressure ratio of approximately 10:1 in the range of 10,000 to 100 Pa. The influence of nozzle wall roughness on the boundary layer characteristics and its subsequent impact on the oblique shock wave behavior, and consequently, on the static pressure distribution along the flow axis, is solved in this paper. It demonstrates the significant effect of varying inertial-to-viscous force ratios at low pressures on the resulting impact of roughness on the oblique shock wave characteristics. The resulting oblique shock wave distribution significantly affects the static pressure profile along the axis, which can substantially influence the scattering and loss of the primary electron beam traversing the differential pumping stage. This, in turn, affects the sharpness of the resulting image. The boundary layer within the nozzle plays a crucial role in determining the overall flow characteristics and indirectly affects beam scattering. This study examines the influence of surface roughness and quality of the manufactured nozzle on the resulting flow behavior. The initial results obtained from experimental measurements using pressure sensors, when compared to CFD simulation results, demonstrate the necessity of accurately setting roughness values in CFD calculations to ensure accurate results. The CFD simulation has been validated against experimental data, enabling further simulations. The research combines physical theory, CFD simulations, advanced experimental sensing techniques, and precision manufacturing technologies for the critical components of the experimental setup.
... In this paper, the governing equation for unsteady flow adopts the N-S equation. In the three-dimensional Cartesian coordinate system it can be written as [20] ...
Traditional elastic correction methods fail to address the significant aeroelastic interactions arising from unsteady flow fields and structural deformations during aggressive maneuvers. To resolve this, a numerical method is developed by solving unsteady aerodynamic equations coupled with a rigid–flexible dynamics equations derived from Lagrangian mechanics in quasi-coordinates. Validation via a flexible pendulum test and AGARD445.6 wing flutter simulations demonstrates excellent agreement with experimental data, confirming the method’s accuracy. Application to a slender air-to-air missile reveals that reducing structural stiffness can destabilize the aircraft, transitioning it from stable to unstable states during forced pitching motions. Studies on longitudinal flight under preset rudder deflection control indicate that the aeroelastic effect increases both the amplitude and period of pitch angles, ultimately resulting in larger equilibrium angles compared to a rigid-body model. The free-flight simulations highlight trajectory deviations due to deformation-induced aerodynamic forces, which emphasizes the necessity of multidisciplinary coupling analysis. The numerical results show that the proposed CFD/CSD-based coupling methodology offers a robust aeroelastic effect analysis tool for flexible flight vehicles during aggressive maneuvers.
... To calculate the turbulence field and, hence, model the Reynolds stresses, which arise in RANS equations after the averaging operation as additional unknowns, the SST k-ω 9 turbulence model [35], which is based on the Boussinesq hypothesis [36], was used. Turbulence transport equations were solved by the finite volume technique with a segregated algorithm [37,38]. For the pressurevelocity coupling, a standard pressure-correction procedure, SIMPLE (semi-implicit method for pressurelinked equations), was applied [39]. ...
Over the past quarter-century, substantial research has been conducted on the potential of dimpled surfaces to passively reduce turbulent friction resistance. This potential is particularly intriguing in the context of external flows, particularly with applications in the transportation sector, such as maritime vessels, rail systems, and aerial vehicles. However, the literature presents conflicting findings and interpretations about the performance of the dimples and the underlying physical mechanisms of the flow. Furthermore, many of the studies do not address the requirements for assessing the efficiency regarding practical engineering, such as high Reynolds number flow and open boundary layer conditions. In this study, the effect of dimpled surfaces on resistance reduction was experimentally investigated. A specialized testing bed, capable of accommodating large test plates, was designed for use in a cavitation tunnel facility. This setup allowed for the achievement of a high Reynolds number range suitable for practical applications, while ensuring that external flow conditions were met. Critical parameters affecting resistance reduction performance such as coverage ratio and boundary layer thickness, were also examined. The resistance values obtained within a broad experimental matrix suggest that, under favourable conditions, dimpled surfaces can be highly effective in terms of energy efficiency. The insights and interpretations drawn from these findings are expected to provide valuable guidance for future research.
... By dividing the computational domain into multiple subregions, this approach allows for more flexible and accurate simulations of intricate geometries and flow patterns (Yu et al., 2002). Additionally, multi-block structured grids offer several advantages in numerical analysis, including decomposing the grid into blocks with simpler topological structures, enabling direct numerical analysis of complex geometries and flows (Blazek, 2015). Each block can be solved independently, making the multi-block method particularly suitable for parallel computing, significantly reducing computational time (Takaki et al., 2003). ...
The finite-difference method (FDM), limited by uniform grids, often encounters severe oversampling in high-velocity regions when applied to multi-scale subsurface structures, leading to reduced computational efficiency. A feasible solution to this issue is the use of non-uniform grids. However, previous discontinuous grid approaches required careful consideration of interpolation operations in transition regions, while single-block continuous grids lacked flexibility. This paper proposes a novel approach using multi-block stretched grids with positive and negative singularities to achieve non-uniform grids, the numerical simulation of seismic waves is realized by combining it with the curvilinear grid finite-difference method (CGFDM). Our method facilitates seamless information exchange between coarse and fine grids without additional interpolation or data processing and allows for flexible grid configurations by adjusting singularity pairs.
The effectiveness of our approach is verified through comparisons with the generalized reflection/transmission method (GRTM) and the finite-element method (FEM). Numerical experiments demonstrate the method's reliable accuracy and significant reduction in grid points compared to uniform grids. Although the stability of our method has not been rigorously mathematically proven, we demonstrate that the algorithm remains applicable for sufficiently long simulations to address realistic scenarios.
... Due to the unsteadiness of the problem, careful solver setting consideration was made. By implementing a second order scheme in time, truncation errors were reduced in comparison to a first order scheme, as the order of the error scales with the time step squared [41]. With a second order implicit unsteady scheme, the simulation was consistently stable for all cases and allowed for a time step to be prescribed. ...
Reverse flow on the retreating side of a rotor disk is an intrinsic aerodynamic limitation of high-speed, high advance ratio rotorcraft. For traditional single main rotor helicopters, the influence of flow reversal is not significant, but it is important for coaxial rigid rotor high-speed helicopters. Flow reversal can be a source of several unsteady flow phenomena such as vortex formation, which increases the pitch link loads that could ultimately lead to fatal crashes. The aim of the current work is to reduce pitch link loads by using blunt trailing-edged blades. Experiments and numerical simulations were compared to a simple low-order model for a quick blade design iteration process. The focus of the present study is on blade aerodynamic loading, namely the blade vertical force, the horizontal force, and the blade pitching moment. A range of advance ratios and blade pitch angles were studied. A 29% pitching moment increase was measured in the reverse flow region with sharp trailing-edged blades compared to blunt blades. The blunt trailing-edged blade delayed flow separation and thus prevented the formation of a reverse flow dynamic stall vortex, reducing the pitching moment. The use of such blunt trailing-edged blades could save pitch links from failing and may ultimately help prevent rotorcraft from fatal crashes.
... In the application of computational fluid dynamics (CFD), the compressible Navier-Stokes equations serve as the governing partial differential equations (PDEs) of compressible viscous flows. Over the past decades, finite volume schemes commonly used for the compressible Navier-Stokes equations primarily employ the method of lines, where the spatial discretizations for the convective and viscous fluxes are treated separately [6]. Typically, the convective flux is discretized based on an exact or approximate Riemann solver, while the viscous flux is treated with a central finite difference method. ...
In the finite volume framework, a Lax-Wendrof type second-order flux solver for the compressible Navier-Stokes equations is proposed by utilizing a hyperbolic relaxation model. The flux solver is developed by applying the generalized Riemann problem (GRP) method to the relaxation model that approximates the compressible Navier-Stokes equations. The GRP-based flux solver includes the effects of source terms in numerical fluxes and treats the stiff source terms implicitly, allowing a CFL condition conventionally used for the Euler equations. The trade-off is to solve linear systems of algebraic equations. The resulting numerical scheme achieves second-order accuracy within a single stage, and the linear systems are solved only once in a time step. The parameters to establish the relaxation model are allowed to be locally determined at each cell interface, improving the adaptability to diverse flow regions. Numerical tests with a wide range of flow problems, from nearly incompressible to supersonic flows with strong shocks, for both inviscid and viscous problems, demonstrate the high resolution of the current second-order scheme.
... While traditional methods have been proven effective in solving PDEs, they still face limitations in terms of computational dimensions and complexity, as well as challenges in terms of computational convergence and inverse problem-solving. This often results in high computational costs (6)(7)(8)(9)(10)12). In recent years, the emergence of artificial intelligence (AI) has brought about a new paradigm for solving PDEs, which has already attracted considerable interest from researchers in mathematics and computational physics (13,14). ...
Modelling complex multiphysics systems governed by nonlinear and strongly coupled partial differential equations (PDEs) is a cornerstone in computational science and engineering. However, it remains a formidable challenge for traditional numerical solvers due to high computational cost, making them impractical for large-scale applications. Neural operators' reliance on data-driven training limits their applicability in real-world scenarios, as data is often scarce or expensive to obtain. Here, we propose a novel paradigm, physics-informed parallel neural operator (PIPNO), a scalable and unsupervised learning framework that enables data-free PDE modelling by leveraging only governing physical laws. The parallel kernel integration design, incorporating ensemble learning, significantly enhances both compatibility and computational efficiency, enabling scalable operator learning for nonlinear and strongly coupled PDEs. PIPNO efficiently captures nonlinear operator mappings across diverse physics, including geotechnical engineering, material science, electromagnetism, quantum mechanics, and fluid dynamics. The proposed method achieves high-fidelity and rapid predictions, outperforming existing operator learning approaches in modelling nonlinear and strongly coupled multiphysics systems. Therefore, PIPNO offers a powerful alternative to conventional solvers, broadening the applicability of neural operators for multiphysics modelling while ensuring efficiency, robustness, and scalability.
... The Eulerian approach divides the spatial region into discrete grids, with fluid properties such as density, velocity, and pressure computed at fixed points. The Lagrangian approach describes the fluid as a system of individual particles, where the motion of each particle is tracked (Blazek 2015). A notable example of the Lagrangian approach is the SPH method, which has been widely adopted for haptic simulations (Cirio et al. 2011a;Wang and Wang 2014). ...
In Virtual Environment (VE), haptic interaction plays an important role in delivering both tactile and kinesthetic sensations, enabling users to perceive the physical properties of virtual objects. These sensory inputs have diverse applications in areas such as medical training, virtual reality (VR) gaming, education, etc. This analytical review aims to provide a comprehensive overview of haptic technologies for fluid interaction developed in the past twenty years. A total of 59 studies meeting the inclusion criteria were identified and examined. The review thoroughly discusses relevant papers on haptic rendering methods as well as haptic devices designed for fluid interaction. In addition, an analytical point of view is presented from four key aspects, including fluid simulation methods, haptic feedback modalities, evaluation approaches, and applications. Finally, this paper highlights the current research gaps and outlines future directions to advance the development of reliable and accurate haptic techniques for interaction with fluids.
... In the last decades, fluid flow simulations have progressively enlarged their applicability and their influence in many different research fields (general overviews can be found in [1][2][3]). Nowadays, applications of computational fluid dynamics (CFD) have reached widespread application areas such as, shape optimization for naval/automotive/aerospace engineering [4,5], cardiovascular in real time surgery [6], chemistry industrial processes [7,8] or weather forecasts [9]. ...
This article provides a reduced-order modelling framework for turbulent compressible flows discretized by the use of finite volume approaches. The basic idea behind this work is the construction of a reduced-order model capable of providing closely accurate solutions with respect to the high fidelity flow fields. Full-order solutions are often obtained through the use of segregated solvers ( solution variables are solved one after another ), employing slightly modified conservation laws so that they can be decoupled and then solved one at a time. Classical reduction architectures, on the contrary, rely on the Galerkin projection of a complete Navier–Stokes system to be projected all at once, causing a mild discrepancy with the high order solutions. This article relies on segregated reduced-order algorithms for the resolution of turbulent and compressible flows in the context of physical and geometrical parameters. At the full-order level turbulence is modeled using an eddy viscosity approach. Since there is a variety of different turbulence models for the approximation of this supplementary viscosity, one of the aims of this work is to provide a reduced-order model which is independent on this selection. This goal is reached by the application of hybrid methods where Navier–Stokes equations are projected in a standard way while the viscosity field is approximated by the use of data-driven interpolation methods or by the evaluation of a properly trained neural network. By exploiting the aforementioned expedients it is possible to predict accurate solutions with respect to the full-order problems characterized by high Reynolds numbers and elevated Mach numbers.
... The spatial approximation is improved by piecewise linear reconstructions with the Barth-Jespersen or Venkatakrishnan limiter [10]. The implicit discretization in time is achieved using the matrix-free lower-upper symmetric Gauss-Seidel (LU-SGS) method [11]. In the case of steady-state problems, the method uses the local time-stepping acceleration method. ...
The contribution proposes a simple virial-like equation of state (EOS) suitable for CFD simulations of real gas flows through turbines or compressors. The virial coe cients are locally optimized using the idealized isentropic flow through the machine. The resulting equation combined with a simple polynomial t for the ideal part of the heat capacity provides a better approximation of the full EOS than standard cubic EOS such as Aungier-Redlich-Kwong.
... Air is considered as an incompressible fluid. The turbulence model is the standard k-model (Blazek, 2015). No-slip condition is assumed at the surface of the condenser. ...
... Analytically solving the aforementioned N-S equations is exceedingly challenging and often infeasible in most scenarios. The conventional approach involves approximating the solutions of these nonlinear PDEs through numerical methods [42], such as the finite element method. In this process, the N-S equations are discretized and reformulated into a solvable system of algebraic equations that effectively characterize the nonlinear fluid dynamics. ...
Conducting repeated high-fidelity simulations of complex turbulent flows entails substantial computational costs in engineering applications. Reduced-order modeling (ROM) seeks to derive low-dimensional representations from full-order numerical systems, thereby facilitating rapid forecasting of future flow states. This study presents a novel data-assisted computational framework that employs deep neural networks for nonlinear ROM of engineering turbulent flows. Specifically, the Stacked Auto-Encoder (SAE) network is utilized for nonlinear dimensionality reduction and feature extraction; the resulting latent features subsequently serve as inputs to the Long Short-Term Memory (LSTM) network for predictive ROM of turbulent fluid dynamics. A comparative analysis is conducted between SAE and proper orthogonal decomposition regarding dimensionality reduction, and the performance of LSTM in time-series forecasting is also evaluated against dynamic mode decomposition, where two different training strategies are applied for LSTM within the reduced-order latent space. The proposed SAE-LSTM-based ROM approach is tested on two typical turbulent flow problems for non-intrusive model order reduction. The results demonstrate that the constructed surrogate models possess significant capability in predicting the evolution of turbulent flows by preserving essential nonlinear characteristics inherent in fluid dynamics. This innovative method shows great promise in addressing computational challenges associated with high-resolution numerical modeling applied to complex large-scale flow problems.
... In addition, tetrahedral mesh elements of appropriate sizes were employed throughout the structure, as shown in Figure 3. This choice of element shape is supported by previous studies [25,43], demonstrating its ability to model complex shapes accurately. The ANSYS analysis settings are listed in Table 6. ...
The development of CubeSats have been advanced following the miniaturization of electronic components. While CubeSats have been extensive used in various missions, most prior research has focused on validating their structural design to comply with deployer requirements. Thus, leaving a gap in understanding the structural performance of different CubeSat frame configurations. This study analyzes two MYSat CubeSat frame designs: modular and monoblock and were conducted using the ANSYS Mechanical package to assess and compare the mechanical properties of each design. The results showed that the modular frame exhibited greater deformation of 4.4 × 10⁻⁷ m and a von Mises stress of 0.74 MPa in the stowed configuration compared to the monoblock. During launch, the modular frame displayed higher deformation and stress under Orbital Cygnus launcher conditions, with natural frequencies of 679.99 Hz and 677.88 Hz for the modular and monoblock frames, respectively. Under PSLV launch conditions, peak stresses were 12.70 MPa for the modular frame and 16.39 MPa for the monoblock frame during random vibration analysis. Stress concentrations were primarily observed at standoffs supporting circuit boards, posing potential risks of loosening or damage. Despite these findings, both designs remained within the Al-6061 yield strength, ensuring structural integrity. This research highlights the importance of evaluating different CubeSat configurations to optimize their design for better resilience under launch stresses and suggests further analysis incorporating larger satellite components to enhance the accuracy of structural performance assessments.
... ∆t s i are the maximum allowable time-steps due to convective, viscous, gravitational and surface tension terms respectively. These time-step restrictions are taken as [33,72,[75][76][77] ...
In this paper, a novel fully-explicit weakly compressible solver is developed for solving incompressible two-phase flows. The two-phase flow is modelled by coupling the general pressure equation, momentum conservation equations and the conservative level set advection equation. A HLLC-type Riemann solver is proposed to evaluate the convective fluxes along with a simple, consistent and oscillation-free discretization for the non-conservative terms. The solver is tested against several two-phase flow problems for its robustness and adaptability on structured as well as unstructured meshes.
... Here, the right hand side g involves spatial derivatives, source terms, etc. as well as the design parameters ρ. Accounting for the unidirectional flow of information in the time domain, an approximation to the unsteady PDE solution is typically evolved forward in time in a step-by-step manner applying a time marching algorithm [4,16,57]. Starting from the initial condition, these schemes march forward in discrete time steps applying nonlinear iterations in space to approximate a pseudo-steady state at each time step as for example in the dual-time stepping approach [33]. However, in many applications, the primal flow is not the only computation of particular interest. ...
In this paper, an adjoint solver for the multigrid in time software library XBraid is presented. XBraid provides a non-intrusive approach for simulating unsteady dynamics on multiple processors while parallelizing not only in space but also in the time domain. It applies an iterative multigrid reduction in time algorithm to existing spatially parallel classical time propagators and computes the unsteady solution parallel in time. Techniques from Automatic Differentiation are used to develop a consistent discrete adjoint solver which provides sensitivity information of output quantities with respect to design parameter changes. The adjoint code runs backwards through the primal XBraid actions and accumulates gradient information parallel in time. It is highly non-intrusive as existing adjoint time propagators can easily be integrated through the adjoint interface. The adjoint code is validated on advection-dominated flow with periodic upstream boundary condition. It provides similar strong scaling results as the primal XBraid solver and offers great potential for speeding up the overall computational costs for sensitivity analysis using multiple processors.
... Tensor trains are widely researched in the context of different numerical problems, including solutions of linear systems of equations [14], optimization [15,16], machine learning [17][18][19], and numerical solutions of PDEs [9,[20][21][22][23][24]. ...
In this paper, we present a methodology for the numerical solving of partial differential equations in 2D geometries with piecewise smooth boundaries via finite element method (FEM) using a Quantized Tensor Train (QTT) format. During the calculations, all the operators and data are assembled and represented in a compressed tensor format. We introduce an efficient assembly procedure of FEM matrices in the QTT format for curvilinear domains. The features of our approach include efficiency in terms of memory consumption and potential expansion to quantum computers. We demonstrate the correctness and advantages of the method by solving a number of problems, including nonlinear incompressible Navier–Stokes flow, in differently shaped domains.
... When applying this turbulence model, two additional transport equations for turbulence kinetic energy and turbulence dissipation should be solved. The RANS and aforementioned turbulence transport equations were solved by finite volume technique with a segregated algorithm (Blazek, 2001;Versteeg and Malalasekera, 2007). A standard pressure-correction procedure, SIMPLE, was applied (Patankar and Spalding, 1972) for the pressure-velocity coupling. ...
Predictions of model tests and numerical studies (Sasaki et al., 2016), both at model-and full-scale, as well as results of the recent sea trials, have demonstrated that the Gate Rudder System (GRS), a novel energy-saving device, increases propulsive efficiency and reduces fuel consumption of a ship. Despite detailed investigations into the impact of the GRS on powering performance and manoeuvring, studies on its effect on cavitation phenomena and underwater radiated noise (URN) are limited. In this study, the effects of the GRS on cavitation and URN were investigated experimentally and compared to a conventional rudder system (CRS). Cavitation observations and URN measurements were conducted on sister ships-one with the CRS, SAKURA, and the other with the GRS, SHIGENOBU-at Istanbul Technical University Cavitation Tunnel (ITUKAT). A wooden ship model was built to be used in resistance and propulsion tests and then truncated due to the cavitation tunnel's test section length limitation of 5.5 m. Numerical studies were performed for both the original and truncated models at model-and full-scale to assess their wake characteristics. The nominal wake distributions along the CRS and GRS propeller planes were measured using a Laser Doppler Velocimetry (LDV) system. The test results supported the findings of the sea trials, showing that the GRS significantly reduces URN levels by up to 10 dB at high-frequencies, compared to the CRS, due to its effect on propeller loading and cavitation phenomenon.
... Despite its widespread use, CFD still faces challenges in areas such as turbulence modelling, combustion, heat transfer, and robust discretization. While advances in computing power have made complex CFD simulations possible on personal computers, further research is needed to address open questions and explore new opportunities in design optimization using CFD [20]. ...
This study investigates the use of titanium dioxide (TiO2) nanofluids to enhance the thermal performance of shell and tube heat exchangers. A comparative computational fluid dynamics (CFD) analysis is conducted using water and a 0.5% TiO2 nanofluid. The heat exchanger is modelled using computer-aided design (CAD), with dimensions closely resembling commercial units. The CFD model is validated through a grid-independence study, with a mesh of 4,112,679 elements yielding grid-independent results. The key findings show that the 0.5% TiO2 nanofluid increases the cold fluid outlet temperature by 11.44% compared to water (36.04°C vs. 33.63°C). The average heat transfer coefficient is enhanced by 12.3% when using the nanofluid. The CFD results are consistent with experimental data, with a maximum deviation of 4.2% in the outlet temperatures. This study demonstrates the successful integration of TiO2 nanofluids with an optimized shell and tube heat exchanger design. The novelty lies in the application of nanofluids to improve the thermal performance of industrial heat exchangers. The presented methodology, combining CAD modelling and CFD analysis, provides a foundation for further optimization and experimental validation of nanofluid-enhanced heat transfer systems.
... The realizable k-ε model, known for its effectiveness in validating separated flows and flows with complex secondary flow characteristics, was selected for turbulence modeling. Its suitability for complex terrains has been verified in various studies [31,36,37]. A double-precision, pressure-based steady-state solver was employed, with the SIMPLEC algorithm used for pressure-velocity coupling. ...
The accurate prediction of the flow field characteristics of complex mountains is of great practical significance for the development and construction of wind farms, but it is not yet fully understood. The main purpose of this study is to propose a method for the study of flow field characteristics under complex mountain conditions, which can optimize the boundary conditions required for numerical simulation through the wind acceleration ratio and, at the same time, couple the numerical simulation and wind measurement data to reflect the real mountain flow field distribution. The results show that the proposed method has good applicability in complex mountain wind farms, can reproduce the real flow field distribution, and has a certain practical value. Wind speed distribution and turbulence intensity are greatly affected by boundary conditions such as wind speed and wind direction and are also affected by the shielding effect brought by terrain changes. The contrast between 120° and 150° wind direction is more obvious. When the incoming wind moves to the top of a mountain or the ridgeline, it will form a low-speed wake area behind it, resulting in reduced wind speed, increased turbulence intensity, and an unstable flow field.
... All flow equations were spatially discretized to be secondorder accurate using a higher-order upwind scheme. The upwind scheme used has been derived by integrating the fluxes over a control volume (Patankar, 1980;Blazek, 2015). Numerical simulation was performed by a Finite Volume Solver, CFD-Ansys Fluent, by solving the transport equationsthe Navier-Stokes Studies in Engineering and Exact Sciences, Curitiba, v.5, n.2, p. 01-20, 2024 equations. ...
Hemodynamic factors play a role in atherogenesis and the localization of atherosclerotic plaques. The human aorta and coronary arteries are susceptible to arterial disease, and there have been many studies of flows in models of these vessels. Atherosclerosis is a systemic disease occurring in specific sections of the cardiovascular tree such as the carotid and the coronary arteries. This paper aims to simulate the human arterial bifurcation and investigates some hemodynamic parameters as blood propagates from the aortic artery toward the carotid artery. We analyze the velocity, static pressure, shear stress. The simulation results showed disturbed flow patterns, such as flow separation and stagnation, as well as abnormal hemodynamic parameters (HPs) distributions including the low and high wall shear stress (WSS) and oscillation of wall shear stress. Based on our simulating characteristic results, the CFD tools can not only monitor the hemodynamic parameters obviously, but also help to analyze the diagnostic for the treatment of vascular diseases. CFD can be an effective technology for the examination and treatment of patients with failure of blood supply of the head and brain.
... Among these, spatially integrating the pressure gradients by means of solving a Poisson equation accompanied by the proper boundary conditions gained popularity due to its superior accuracy (Charonko et al 2010). Furthermore, Neeteson and Rival (2015) proposed a novel solution approach of utilizing Poisson equation for integration of pressure gradients on scattered Lagrangian domains by constructing a computational grid using Voronoi tessellation (VOR) (Hirata 2005) and Delaunay triangulation (Blazek 2005). However, a comparative assessment of various pressure reconstruction techniques (van Gent et al 2017), demonstrated that the solution of the Poisson equation on gridded data yields the minimum pressure reconstruction errors with respect to a known ground truth. ...
The coupling between fluid-structure interactions is governed by the pressure distribution over the interaction surface between the fluid and solid domains. The capabilities of non-intrusive optical techniques, such as particle image velocimetry and Lagrangian particle tracking (LPT), have been proven to provide accurate velocity and acceleration information within the flow field while simultaneously tracking the corresponding structural deformations. However, scattered data from LPT measurements are typically mapped onto Cartesian grids, independently of the shape of the solid objects in the measurement domain. The use of Cartesian grids poses challenges for the determination of the surface pressure because the velocity gradients close to the object’s surface are not captured accurately. Therefore, an alternative surface pressure reconstruction scheme utilizing LPT data based on the arbitrary Lagrangian–Eulerian approach is proposed to mitigate the error propagation associated with the use of uniform grids. The introduced method provides an exact surface conformation utilizing boundary fitted coordinate systems and radial basis function based mesh deformations, which eliminates the need to use extrapolations to obtain surface pressure distributions. The introduced approach is assessed by means of a synthetic hill surface probing a three-dimensional analytical flow field; its practical applicability is demonstrated through an experimental characterization of turbulent boundary layer interactions with a steadily and unsteadily deforming elastic membrane.
... The CFX solver of Ansys discretizes the turbulence model using a cell-vertex scheme [48]. Therefore, the mathematical definition of the rotational periodic boundary conditions is defined as [54]: ...
Hydraulic turbines have become indispensable for harnessing renewable energy sources, particularly in-pipe hydraulic turbine technology, which leverages excess energy within pipeline systems like drinking water distribution pipes to produce electrical power. Among these turbines, the propeller-type axial turbine with circular blades stands out for its efficiency. However, there is a notable lack of literature on fluid dynamics and structural behavior under various operational conditions. This study introduces a comprehensive methodology to numerically investigate the hydraulic and structural responses of turbines designed for in-pipe installation. The methodology encompasses the design of circular blades, followed by parametric studies on fluid dynamics and structural analysis. The circular blade’s performance was evaluated across different materials, incorporating static, modal, and harmonic response analyses. Results showed that the circular blade achieved a peak hydraulic efficiency of 75.5% at a flow rate of 10 l/s, generating 1.86 m of head pressure drop and 138 W of mechanical power. Structurally, it demonstrated a safety factor exceeding 1 across the entire hydraulic range without encountering resonance or fatigue issues. This research and its methodology significantly contribute to advancing the understanding of designing and assessing the fluid dynamic behavior and structural integrity of circular blades in axial propeller-type turbines for in-pipe installations, serving as a valuable resource for future studies in similar domains.
... Computational Fluid Dynamics (CFD) is an example of a numerical simulation method that has recently been adopted to solve complex physics involved in the simulation of additive manufacturing process [21]. The method involves simulating fluid flow by solving momentum and mass conservation equations and simulating fluid interaction with solid bodies [22]. CFD is particularly useful in PBF as it allows parameter changes to be easily varied over wide ranges such as input power, scan strategies and powder characteristics, facilitating design optimisations [23]. ...
This study investigates the influence of processing parameters, specifically laser power and scan speed, on the melt pool characteristics of Ti-6Al-4V alloy during Powder Bed Fusion–Laser Beam (PBF-LB) printing. The objective is to provide a deeper understanding of the melt pool based on in-process melt pool monitoring data obtained from a production scale PBF-LB system (RenAM 500 S), experimental melt pool characterisation from single line print studies, and numerical modelling studies using Computational Fluid Dynamics (CFD). The findings demonstrate that laser power has a more pronounced effect on melt pool depth compared to scan speed, for a given line scan energy. This is due to an increased dwell time at elevated temperatures and reduced local cooling rates. A direct correlation is established between melt pool depth, increased emission intensity from in situ monitoring system, and the temperature-time profile derived from numerical modelling. Additionally, melt pool fluid flow based on numerical modelling reveals the presence of intense thermocapillary flow at higher laser power conditions, leading to the retention of porosity along the track. This observation was supported through experimental validation, as evidenced by the increased levels of porosity observed in line scan samples printed at higher laser powers.
... Analytically solving the above N-S equations is extremely difficult and usually impossible in most cases. The routine way is to approximate the solution of non-linear PDEs using numerical approaches [61], such as the finite element method. The N-S equations are discretized and reformulated into a solvable system of algebraic equations that approximately characterise the nonlinear fluid dynamics. ...
Repetitively conducting high-fidelity numerical simulations under varying conditions is often a crucial requirement in the optimization design of offshore bridges and structures. Reduced-order modeling (ROM) provides an efficient approach to quickly and reliably obtain solutions by extracting low-dimensional representations from full-order numerical systems. This paper presents a novel data-driven computational framework for non-intrusive ROM of turbulent/unsteady flows passing around bridge piers, consisting of two interconnected components: the Stacked Autoencoder (SAE) and the Dynamic Mode Decomposition (DMD). The novelty lies in utilizing SAE to achieve nonlinear dimensionality reduction by projecting the full-order dynamical system onto a low-dimensional latent space, followed by constructing reduced-order models through data-driven DMD to represent fluid dynamics in the latent feature space. This new SAE-DMD-based method is applied to develop reduced-order models for two unsteady flow problems, and it is also compared with classical DMD and high-fidelity numerical simulations in terms of modeling accuracy, forecasting efficiency and memory requirements. The results demonstrate that the proposed method can rapidly offer reliable predictions while significantly reducing memory usage and it exhibits excellent extrapolation capability by accurately preserving primary nonlinear characteristics of fluid dynamics. This new method shows potential to overcome computational challenges associated with high-resolution numerical modeling for complex large-scale flow problems.
... The most common flow regime is turbulent, which is characterized by the presence of three types of motion -translational, rotational and oscillatory. Currently, mathematical models are used, implemented in the form of a combination of exact and semi-empirical equations that do not take into account rotational motion [1][2][3][4]. In this paper, problems are considered within the framework of an averaged turbulence model, which takes into account the influence of translational and rotational motion [1,2,4]. ...
Introduction Turbulent and laminar flows have a great influence on natural processes, as well as on energy and transport. To effectively use the capabilities of these processes, it is necessary to have the most accurate mathematical description of the movement of continuous media. This description is based on the law of conservation of momentum and uses partial differential equations. The most common flow regime is turbulent, which is characterized by the presence of three types of motion-translational, rotational and oscillatory. Currently, mathematical models are used, implemented in the form of a combination of exact and semi-empirical equations that do not take into account rotational motion [1-4]. In this paper, problems are considered within the framework of an averaged turbulence model, which takes into account the influence of translational and rotational motion [1, 2, 4]. The laws of motion of working bodies as a continuous medium are studied in fluid mechanics, the mathematical basis of which is the equations of motion in stresses (Navier). In the classical literature, there are two exact three-dimensional special cases of these equations: the equation of motion for a rigid body (elasticity theory) and the Navier-Stokes equation for a liquid. The theory of elasticity has proven the high quality of calculations and has a clear structure of equations [5, 6]. In fluid mechanics there is no similar structure for all flow regimes, which led to a large share of experimentation and, accordingly, to a high labor intensity of research. Writing accurate equations requires accepting the point of view that the general equation of motion must describe the most general (turbulent) flow regime. The implementation of this point of view became possible by applying the operation of isolating the velocity rotor from the expressions for strain rates and from the Laplace operator of velocity. In this case, the second form of the equation was used for the total acceleration of a liquid particle in the Gromeka-Lamb form, which includes the angular velocity of rotation of the particles [4].
... F c and F v are the convective flux and viscous flux, respectively. Their specific forms can be found in [20]. ...
This study proposes a computational fluid dynamics and computational structure dynamics (CFD/CSD) coupled method for calculating the buffet response of a variant tail wing. The large-scale separated flow in the buffet is simulated by the detached vortex approach, vibration deformation of the tail wing is solved by the dynamic mesh generation technique, and structural modeling is based on the mode method. The aerodynamic elastic coupling is calculated through the cyclic iteration of aerodynamics and the structural solution in the time domain. We verify the correctness of the proposed method through a typical delta wing calculation case, further simulate the buffet response of a variant tail wing in maneuver state, and finally realize buffet mitigation using an active excitation method. Overall, this study can provide an important reference for the design of variant-tailed aircraft.
The present work primarily focuses on the study of three gradient reconstruction techniques applied to the calculation of viscous terms in a cell-centered, finite volume formulation for general unstructured grids. The work also addresses different ways of formulating the limiter functions necessary to maintain stability in the presence of flow discontinuities. The flows of interest are simulated using the compressible Reynolds-averaged Navier–Stokes equations, and the negative Spalart–Allmaras model is used for turbulence closure. Definition of interface inviscid terms uses the Roe approximate Riemann solver, whereas the interface viscous terms are calculated with a standard centered scheme together with appropriate definitions of the interface gradients. Steady-state solutions are obtained using an implicit time-integration method, together with a novel convergence acceleration technique. This new approach defines a set of three simple rules for controlling the global CFL number based on the residue evolution. The work considers three test cases, namely the subsonic bump-in-channel flow, the subsonic NASA high-lift common research model multielement airfoil and the transonic ONERA M6 wing. Present results are compared to experimental and numerical data available in the literature. Severe numerical instabilities are observed when the simplest gradient reconstruction technique is used, while more sophisticated formulations are able to provide excellent agreement with the existing literature. Current results are demonstrated to be highly insensitive to modifications made to the entropy fix terms of the numerical flux. Integrated aerodynamic forces are shown to be mildly dependent on the limiter formulation used, even in the absence of shock waves. The proposed convergence acceleration procedure manages to quickly drive the residue terms to machine zero, provided no major instabilities are present.
This study focuses on 2D RANS (Reynolds Averaged Navier-Stokes) simulations using Spalart-Allmaras and k- SST turbulence models for a supersonic air inlet featuring two different passive control systems: an air bleed system in the external ramp of the inlet and a two-dimensional bump. The supersonic inlet serving to capture and decelerate the high-speed incoming flows is aerodynamically indispensable to an airbreathing supersonic aircraft. Sometimes, depending on the conditions of the entry flow, the shock wave boundary layer interaction (SWBLI) can lead to inlet unstart if not controlled, due to thickened boundary layer. To verify the impact of the passive control systems, the inlet was tested at freestream Mach number of 2.0 and 2.03 as the geometry is very sensitive to Mach number change. Results indicate that the air bleed system is more effective for Mach 2.0 and reduces the bubble size of approximately 80.0%. In the case of the two-dimensional bump, it was noticed that the bump should be placed after the impinging shock on the geometry. Even though the bubble size does not reduce as much as for the air bleed system, for the two-dimensional bump, the SWBLI is weakened.
Marine propeller design requirements have risen in quantity and quality in recent decades. Reduced propeller cavitation is targeted to ensure that comfort requirements and environmental regulations are met. This paper presents the development of a mesh refinement process for the numerical prediction of tip vortex cavitation (TVC) using the commercial CFD package STAR-CCM+. Given the strong dependence on the mesh resolution within the areas of interest, mesh refinement and the use of field functions for adaptive meshing were demonstrated. The developed numerical model was substantiated against relevant published test data. Subsequently, the validated mesh refinement process was extended to scaled-up models representing medium- and full-scale propellers. The results showed that this process can be applied to CFD simulations to capture the minimum pressure within a tip vortex core. This process is also applicable to different types of hydrodynamic propulsors at both model scale and full scale. Additionally, the cavitation inception scaling law was evaluated for all small-scale and full-scale models, and it was found that the scaling parameter obtained using the developed refinement process was somewhat close to that obtained using existing methods. It is expected that the mesh refinement process developed in this study can be used to investigate the effect of scaling on tip vortex cavitation inception.
This study aims to investigate the effect of forward speeds on the flow structure and dynamic stall events of a single rotating blade with pitching motion. Two flight speeds are examined: moderate (µ = 0.3) and high (µ = 0.35) forward flight speeds. To study the three-dimensional flow field, unsteady Reynolds averaged Navier–Stokes (URANS) equations are solved using finite volume discretization and kω-SST turbulence modeling. The present simulations are carefully validated and verified by comparing their results with the experimental data of Caradonna–Tung rotor blade at transonic hover flight and the AH1-G helicopter blade at its maximum forward flight speed. First, the changes in aerodynamic loads and the evolution of various vortex structures on the r/R = 0.778 radial section during the dynamic stall cycle are investigated. Our findings indicate that the stall occurred on the advancing side of both flight speeds due to a shock-induced separation. Development of the stall on the retreating side began with the formation of turbulent separation vortices (TSVs) at the trailing edge region, which then evolved into dynamic stall vortices (DSVs), resulting in the occurrence of multiple dynamic stalls at moderate forward speed. At high forward speed, however, one major stall occurs eventually. Next, the complex three-dimensional flow pattern that emerges on the blade’s surface is investigated. This examination illustrates the accumulation of DSVs and the formation of a spanwise vortical structure during pitch-up motion. Also, streamwise vortices (DS-like) are formed on the blade in the pitch-down motion. These phenomena appear at both forward flight speeds. Moreover, simulation findings reveal the dominance of radial flow during the post-stall stage, particularly during pitch-down motion at both flight speeds, significantly affecting vortex evolution at the radial locations of the blade.
The paper examines computational schemes for calculating the gradient of fluid dynamic quantities using grids of various types. The Green–Gauss method and the least squares method (LSM) used to develop a hybrid gradient calculation scheme are considered. It is demonstrated that the accuracy of gradient calculations may vary depending on the geometry of the control volume: the Green–Gauss method exhibits lower errors for strongly elongated thin cells and cells with curved edges, while for cells with orthogonal edges, it is preferable to use LSM. In order to improve the accuracy of calculations on unstructured grids, a hybrid gradient calculation scheme is proposed. This scheme computes the gradient by summing values derived from both the Green–Gauss method and LSM, given the weight function that incorporates the geometry of the control volume. The paper presents a formula for the weight function, which determines the contribution of each method within the hybrid scheme. The developed scheme is applied to the problem of supersonic flow around a cylinder with a needle on two unstructured grids, namely truncated hexagons and tetrahedra. It is shown that the proposed hybrid scheme reduces the error in calculating the aerodynamic characteristics of a streamlined object.
The analysis of high-speed flow is an area that is growing increasingly due to current technologies, it is essential to develop tools and methods to address, precisely such physical phenomena. A major tool in this process is the computational analysis of fluid dynamics, this implementation that is based on the equations that govern the physical phenomenon using numerical methods. This work, by HYNE2D source code, aims to conduct numerical simulations in a flow at high speed through some geometries in order to analyze the behavior of the properties along the flow for each case. Finally, we compared these properties in order to analyze the influences that each profile has when undergoing the same conditions, starting a method of profile selection that will be used in future by the staff of the Federal Technological University of Paraná rockets - Campus Pato White. The source code used in its implementation has the discretization method of finite volumes, the solution domain through unstructured meshes and the Runge-Kutta method for time advance. The case simulated here are models of laminar flows, considering the atmospheric air model "frozen", governed by the Navier-Stokes equations.
Due to the complex nature of the built environment, urban wind flow is unpredictable and characterised by high levels of turbulence and low mean wind speed. Yet, there is a potential for harnessing urban wind power by carefully integrating wind turbines within the built environment at the optimum locations. This requires a thorough investigation of wind resources to use the suitable wind turbine technology at the correct location—thus, the need for an accurate assessment of wind resources at the proposed site. This paper reviews the commonly used wind assessment tools for the urban wind flow to identify the optimum tool to be used prior to integrating wind turbines in urban areas. In situ measurements, wind tunnel tests, and CFD simulations are analysed and reviewed through their advantages and disadvantages in assessing urban wind flows. The literature shows that CFD simulations are favoured over other most commonly used tools because the tool is relatively easier to use, more efficient in comparing alternative design solutions, and can effectively communicate data visually. The paper concludes with recommendations on best practice guidelines for using CFD simulation in assessing the wind flow within the built environment and emphasises the importance of validating CFD simulation results by other available tools to avoid any associated uncertainties.
近年来，由于计算机软硬件及程序设计的飞速发展，使得数值仿真与理论分析和试验研究一同，逐渐成为了一种研究流体力学的重要方法。但随着科研与实际应用中的所面临问题的日趋复杂以及对流动模拟的精细化要求，现有的计算机硬件始终无法适应不断增长的计算需求。如何有效地提升流场求解速度，开发出快速的迭代算法，仍是各学科研究的重点与紧迫课题。
为了克服算法的准确性与效率的之间的冲突，学者们开发了多种先进迭代算法和加速算法来加快算法的收敛速度，例如：计算流体动力学的多重网格算法、矩阵的超松驰迭代算法、最小残差算法、深层神经网络的学习算法等。但是，这些加速算法通常都不具备普适性，并且依赖于用户的实际经验，所以研究高效、鲁棒的加速收敛方法是一项非常有价值的研究课题。
本文对加速收敛方法进行广泛调研，并主要开展了如下工作：
（1）归纳总结了以往加速收敛方法的思路，并依据其加速原理进行了分类。数据驱动的加速收敛方法，作为一类新型加速方法，充分利用了迭代中的过程信息来实现加速收敛。在此框架下，对现有加速方法进行了分类和整理，并评述了各类方法之间的优缺点。
（2）借鉴了降阶模型和最小残差类方法的思路，创新性地提出了基于均值的最小残差方法用于加速流场求解。多个典型算例表明基于均值的最小残差方法加速效果明显，并且对参数不敏感，鲁棒性好。对于隐式伪时间迭代方法加速比达到2-7倍。此外，将基于均值的最小残差方法与其它加速方法进行了对比，从理论上分析比较了不同方法之间的差异，分别从投影方法和拟牛顿迭代方法角度进一步阐释了该方法的原理，并证明了其对线性迭代算法的加速效果。
（3）探究了基于均值的最小残差方法以及模态多重网格在加速收敛以外的应用，包括增强伪时间迭代算法的收敛性和稳定性、求解不稳定定常解以及加速收敛大型线性方程组迭代。结果表明，数据驱动的加速收敛方法能够显著增强迭代算法的稳定性。此外，加速后的雅可比迭代能超越广义最小残差方法。
（4）在模态多重网格方法的基础上，融入寻优算法。通过以最小残差为目标函数，优化动力学模态的系数，能够进一步提升模态多重网格方法的加速效果，解决了传统模态多重网格对渐进收敛模态失效的问题。典型算例表明，加入优化算法后，迭代后期的加速效果明显提升，加速效果更为明显。
作为数据驱动的加速收敛方法，本文提出的基于均值的最小残差方法以及优化模态多重网格方法，初步为数值算法提供了一种通用的加速策略，这种策略对于场加速收敛、提升迭代算法稳定性和收敛性、求解不稳定定常解、加速求解线性方程组等均取得了显著效果。此外，模态多重网格在提升数值稳定性和收敛性方面具有显著效果，基于优化算法的模态多重网格方法也展现出了极大的加速收敛潜力。
The study investigates the transient dynamics of a third-grade fluid, capable of undergoing exothermic reactions, in a two-dimensional rectangular micro-channel. The combined effects of the adverse pressure gradients and electro-osmotic forces constitute the primary flow drivers. In addition to exothermic reactions, the system if also subjected to joule heating and convective cooling at the micro-channel boundaries. Newton’s law of cooling and Arrhenius kinetics are employed to model the boundary-cooling and exothermic-reactions respectively. The temperature-dependent fluid viscosity is modelled via a Nahme-type law. It is assumed that the area in between the micro-channels is a porous material with constant permeability. Computational solutions (implemented on the MATLAB software) are employed for the non-homogeneous partial differential equations for temperature and velocity. These computational solutions are developed from efficient, convergent, and unconditionally stable, semi-implicit finite difference methods. In contrast, the linearized Poisson–Boltzmann equation is solved analytically. The sensitivity of the field variables to variations in the various flow parameters are explored graphically and discussed qualitatively.
An explicit Navier-Stokes solver has been written with the option of using one of two types of turbulence model. One is the Baldwin-Lomax algebraic model and the other is an implicit k-[var epsilon] model which has been coupled with the explicit Navier-Stokes solver in a novel way. This type of coupling, which uses two different solution methods, is unique and combines the overall robustness of the implicit k-[var epsilon] solver with the simplicity of the explicit solver. The resulting code has been applied to the solution of the flow in a transonic fan rotor, which has been experimentally investigated by Wennerstrom. Five separate solutions, each identical except for the turbulence modeling details, have been obtained and compared with the experimental results. The five different turbulence models run were: the standard Baldwin-Lomax model both with and without wall functions, the Baldwin-Lomax model with modified constants and wall functions, a standard k-[var epsilon] model, and an extended k-[var epsilon] model, which accounts for multiple time scales by adding an extra term to the dissipation equation. In general, as the model includes more of the physics, the computed shock position becomes closer to the experimental results.