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A Level Set Formulation of Eulerian Interface Capturing Methods for Incompressible Fluid Flows
Abstract
A level set formulation is derived for incompressible, immiscible Navier–Stokes equations separated by a free surface. The interface is identified as the zero level set of a smooth function. Eulerian finite difference methods based on this level set formulation are proposed. These methods are robust and efficient and are capable of computing interface singularities such as merging and reconnection. Numerical experiments are presented to demonstrate the effectiveness of the methods.
... However, the interactions of the two-phases are directly captured by a fractional step approach, where the pressure Poisson equation is recast using constant coefficients building upon the ideas in [14] and [13]. A conservative level-set approach is used based on the dual-redistancing scheme proposed in [17,18] for mass conservation, while the pressure jumps at the interface are enforced using the Ghost Fluid Method (GFM) [19]. While the previous approaches are not novel in themselves, to the author's knowledge it is the first time they are combined in such a formulation. ...
... The level-set approach has been widely used in modeling two-phase flows from its relative simplicity through the use of a scalar function ϕ classically defined as a signed distance function [17,21,22], ...
... Numerical diffusion can lead to mass imbalance, where errors are prone to increase with local curvature due to grid resolution. To enforce global mass conservation for each phase, a second redistancing procedure is employed [17,18]. The perturbed Hamilton-Jacobi equation is marched toward steady-state: ...
Strong turbulence interacting with a free-surface is a challenging problem, where the interface between gas and liquid is disrupted, generating droplets and bubbles. Turbulent bubble entrainment and breakdown is of paramount importance to quantify mass and momentum transport, and developing lower order models. Thus far, computational costs of scale resolving simulations (for both interface and turbulence) have limited the ability to study these flows. In this work we leverage recent advances in two-phase solvers, including a splitting scheme for the Poisson equation allowing the use of FFT-based Fast Poisson Solvers (FPS) greatly reducing solution time, with verifiable accuracy; coupled with the Ghost Fluid Method (GFM) for a sharp interface treatment. The proposed formulation uses the level set approach, with a dual-redistancing scheme for global mass conservation of each phase. To analyze strong free-surface turbulence, homogeneous isotropic turbulence (HIT) is forced under an air-water interface to analyze stationary statistics of the flow including entrained bubble sizes and Reynolds stresses. Two different approaches are used for the sub-surface forcing. The first uses the common linear forcing scheme in physical space, which imposes a constraint on the domain size. The alternative approach relies on the synthetic random Fourier method to generate a boundary condition which reduces the total computational domain, allowing for higher Reynolds number simulations of turbulent free-surface flows.
... This determines the evolution of the lamellipodium shape through the solution of a Hamilton-Jacobi equation, coupled to the elastic equilibrium equation. The resulting mathematical problem is amenable to numerical simulation via the level set method (29,30) which has been applied to cell evolution study (17,31). In addition to the substrate stress field, the evolving shape of the lamellipodium is the main output of the model. ...
... We use the level set method (29,30) which has been successfully applied to cell evolution study, e.g., (17,31) to solve for the evolution of the lamellipodium boundary C t together with the other model equations. The level set function ϕ(x, t) vanishes on C t , is positive inside Ω t and negative outside it. ...
... This procedure is crucial for our formulation, since the extension of the normal velocity V in our case is not continuous across the phase boundary in the sharp-interface ε limit. This makes computations more difficult than in the fluid interface problem considered in (30,38), where the normal velocity is continuous across the interface. ...
A mathematical model is proposed for shape evolution and locomotion of fish epidermal keratocytes on elastic substrates. The model is based on mechanosensing concepts: cells apply contractile forces onto the elastic substrate, while cell shape evolution depends locally on the substrate stress generated by themselves or external mechanical stimuli acting on the substrate. We use the level set method to study the behavior of the model numerically, and predict a number of distinct phenomena observed in experiments, such as (i) symmetry breaking from the stationary centrosymmetric to the well-known steadily propagating crescent shape, (ii) asymmetric bipedal oscillations and traveling waves in the lamellipodium leading edge (iii) response to mechanical stress externally applied to the substrate (tensotaxis), (iv) changing direction of motion towards an interface with a rigid substrate (durotaxis) and (v) the configuration of substrate wrinkles induced by contractile forces applied by the keratocyte.
... A wide variety of techniques have been proposed to model multiphase fluid systems with incompressible, immiscible fluids. These include explicit front-tracking techniques [1,2], the volume-of-fluid method [3,4], the phasefield method [5,6,7], and the level set method [8,9]. A constant challenge in each of these techniques is the conservation of mass during the course of the simulation. ...
... Some techniques, such as the Immersed Interface Method [21], augment the discretization of the differential equations to take into account the jumps across the interface. Another technique, which is explained here, is to model the domain as a "single" fluid with spatially varying properties [9,22]. The interface condition, Eq. ...
... Additional information regarding the semi-Lagrangian method for Navier-Stokes equations is given in Ref [27]. The next step of a standard projection method would be to determine the pressure to satisfy local volume conservation, Eq. (9). In this work global conservation is also considered. ...
The conservation of mass is common issue with multiphase fluid simulations. In this work a novel projection method is presented which conserves mass both locally and globally. The fluid pressure is augmented with a time-varying component which accounts for any global mass change. The resulting system of equations is solved using an efficient Schur-complement method. Using the proposed method four numerical examples are performed: the evolution of a static bubble, the rise of a bubble, the breakup of a thin fluid thread, and the extension of a droplet in shear flow. The method is capable of conserving the mass even in situations with morphological changes such as droplet breakup.
... The family of boundary unfitted approaches, on the other hand, considers meshes which are not fitting with the interface. Several approaches fall in this family: we mention, for instance, the level set formulation [19], the Nitsche-XFEM method [1,18], the fictitious domain approach introduced in [25,26], the cut-FEM method [17], the shifted boundary method [5] and the immersed boundary-conformal isogeometric method [37]. In our case, the fluid and solid are discretized by two completely independent meshes and then the solid discretization is in some way superimposed to the fluid one. ...
... and the continuous one (u, p, X, λ). The error analysis relies again on inf-sup conditions similar to (19). Since the bilinear form B is computed exactly, we assume that the discrete form A h satisfied the inf-sup condition; the proof is postponed to Section 8. ...
... Thanks to the results in [38], Assumption 1 and the first inf-sup condition in (19) [16] imply the following inf-sup condition [6] (29) ...
We consider a fictitious domain formulation for fluid-structure interaction problems based on a distributed Lagrange multiplier to couple the fluid and solid behaviors. How to deal with the coupling term is crucial since the construction of the associated finite element matrix requires the integration of functions defined over non-matching grids: the exact computation can be performed by intersecting the involved meshes, whereas an approximate coupling matrix can be evaluated on the original meshes by introducing a quadrature error. The purpose of this paper is twofold: we prove that the discrete problem is well-posed also when the coupling term is constructed in approximate way and we discuss quadrature error estimates over non-matching grids.
... For the fluid-fluid interface capturing, some numerical methods were developed to investigate interfacial dynamics such as the front-tracking method [1], level set (LS) method [2], volume of fluid (VOF) method [3], and diffuse interface method [4]. Due to an artificial interface rupture based upon some ad hoc criteria [1], the front-tracking method is not * szh070318@zufe.edu.cn ...
... Once the hydrodynamic force F h is obtained, the hydrodynamic torque T h can be computed, and then the translational and angular velocities are updated by Eqs. (2) and (3). One merit of SPM is the calculation of the hydrodynamic interaction force F h on the particle as one of the direct force methods, which can be obtained by the summation of the fluid-solid interaction force f p at the position x. ...
... In the simulation, the computational domain is divided into a 256 × 256 grid with the same boundary conditions as the previous case. The properties of the fluids and cylindrical particles are given by ρ l = ρ g = 1, η l /η g = 1.0, and (ρ p1 , ρ p2 ) = (1.5, 1.5), (0.5, 0.5), and (1.5, 0.5) for cases (1), (2), and (3), respectively, and Bond number |Bo| = 0.257 ...
In this paper a phase-field based lattice Boltzmann equation (LBE) is developed to simulate wettable particles fluid dynamics together with the smoothed-profile method (SPM). In this model the evolution of a fluid-fluid interface is captured by the conservative Allen-Cahn equation (CACE) LBE, and the flow field is solved by a classical incompressible LBE. The solid particle is represent by SPM, and the fluid-solid interaction force is calculated by direct force method. Some benchmark tests including a single wettable particle trapped at the fluid-fluid interface without gravity, capillary interactions between two wettable particles under gravity, and sinking of a horizontal cylinder through an air-water interface are carried out to validate present CACE LBE for fluid-fluid-solid flows. Raft sinking of multiple horizontal cylinders (up to five cylinders) through an air-water interface is further investigated with the present CACE LBE, and a nontrivial dynamics with an unusual nonmonotonic motion of the multiple cylinders is observed in the vertical plane. Numerical results show that the predictions by the present LBE are in good agreement with theoretical solutions and experimental data.
... This comes at the price of reducing the accuracy of the scheme unless special techniques are used. Within this family of methods, we mention, for instance, the level set method [42,24], the immersed boundary method [40,41], the fictitious domain approach [34,33], the immersed boundary-conformal isogeometric method [44], the shifted boundary method [39,6], the virtual element method on polygonal pixel-based tessellations [11], the fat boundary method [10] and, finally, the Nitsche-XFEM method [1,19]. ...
... We set c h = c 0,h . The continuity of c 0,h (µ h , v h (X)) is a direct consequence of the continuity of c 0 combined with (24) and the inclusion X(B) ⊂ Ω; indeed it holds ...
We consider a distributed Lagrange multiplier formulation for fluid-structure interaction problems in the spirit of the fictitious domain approach. Our previous studies showed that the formulation is unconditionally stable in time and that its mixed finite element discretization is well-posed. In this paper, we analyze the behavior of the condition number with respect to mesh refinement. Moreover, we observe that our formulation does not need any stabilization term in presence of small cut cells and conditioning is not affected by the interface position.
... Our mass transfer model was described in Welch and Wilson [25] in connection with a Volume of Fluid (VOF) implementation. The advection equation for the level set function is solved using the essentially non-oscillatory (ENO) scheme [32], while the advection of the VOF function is achieved through the geometric advection method. Once the location of the interface is identified in the computational domain, we use the embedded boundary method [16] to accurately capture the phase change process at the interface. ...
... We derive the pressure Poisson equation from the augmented continuity equation and the momentum equation, which is solved iteratively by preconditioned conjugate gradient scheme of Van der Vorst [34]. The advection terms of the momentum equations are discretized using the ENO scheme [32], while the diffusive terms are discretized using the central difference scheme. The detailed formulation and numerical method utilized in this study can also be found in Ref. [14]. ...
The evaporation of a liquid drop of initial diameter (Ddrop) migrating in a tube of diameter (D0) is investigated using the coupled level set and volume of fluid (CLSVOF) method focusing on determining the heat and mass transfer coefficients for a deforming drop. A robust phase change model is developed using an embedded boundary method under a finite difference framework to handle vaporizing flows. The model is extensively validated through simulations of benchmark problems such as arbitrary evaporation of a static drop and reproduction of psychrometric data. The results show that the Sherwood number (Sh) and the Nusselt number (Nu) reach a steady value after an initial transient period for the drop subjected to Hagen-Poiseuille flow. A parametric study is conducted to investigate the effect of drop deformation on the rate of evaporation. It is observed that Stefan flow due to evaporation has a negligible impact on the drop deformation dynamics. We also observed that, for different values of Ddrop/D0, the Sh follows a linear correlation with Re^{1/2}Sc^{1/3}.
... In this case, meshes are generated without taking care of the interface, which is allowed to cross elements. We mention, for instance, the Immersed Boundary Method [66,67], where the interface position is tracked by Dirac delta functions, and the level set method [71,34], where the interface corresponds to the zero level set of a certain function. Other techniques such as Nitsche-XFEM [1], Cut-FEM [28,31,52], Finite Cell methods [38,39,48], and Ghost-FEM [7] are based on the use of penalty terms. ...
We review the main features of an unfitted finite element method for interface and fluid-structure interaction problems based on a distributed Lagrange multiplier in the spirit of the fictitious domain approach. We recall our theoretical findings concerning well-posedeness, stability, and convergence of the numerical schemes, and discuss the related computational challenges. In the case of elliptic interface problems, we also present a posteriori error estimates.
... In such type of method, scalars (q; a; /) are located at the cell centers, and velocity components are assigned at the center of the cell faces. The convection and the viscous terms were discretized by a second order ENO method 61 and central differencing, respectively. An explicit time stepping method was adopted in the present study, and to maintain the stability of the solution, time steps are chosen to satisfy Courant-Friedrichs-Lewy (CFL), capillary, viscous, and gravitational time conditions. ...
Electrohydrodynamics (EHD) deals with the motion of a liquid under the influence of an external electric field, and a leaky dielectric fluid has low but non-zero conductivity. Many earlier studies on EHD have considered the flow of perfect dielectric fluids. This study has established a computational framework to investigate the droplet evolution pattern of leaky dielectric fluids under an externally applied electric field. The Navier–Stokes equations and charge conservation equations are solved using the coupled level set and volume of fluid method. It was observed that the dynamics of the leaky dielectric liquid emanating from the orifice of an ejector depends on several factors such as flow rate, electric field intensity, surface tension, Coulombic forces, and electrophoretic forces. Different jetting regimes, such as choked jets, pulsating jets, steady cone jets, and multi jets, were observed based on these conditions. Variations in the drop diameters and jet breakup length at different flow rates were studied, and the results were compared with the available experimental results. It was also observed that the breakup time and detached drop volume can be suitably tuned by varying the strength of the applied electric field.
... The advection equation for the level-set is discretized in space using a fifth-order weighted essentially nonoscillatory scheme [23] and advanced in time with a third-order total variation diminishing Runge-Kutta scheme [24]. A dual-redistancing approach is used at every time step to ensure the level-set remains a signed distance function while maintaining the global mass conservation of each phase [25,26]. Numerical tests show that this dual-redistancing approach works very well and achieves low errors in mass conservation while avoiding the additional complexities of other approaches such as coupled volume of fluid (CVOF) [22]. ...
Two-phase flows involving the dynamic interaction of gas and liquid phases are fundamental to both natural and industrial systems. A critical phenomenon in these flows is bubble fragmentation, which affects interfacial area and mass/momentum transfer. Direct numerical simulations (DNS) of turbulent two-phase bubbly flows allow for improved control of physical parameters and access to local flow variables that are challenging to obtain from traditional experiments. In this work, we perform numerical experiments of bubble breakup in decaying isotropic turbulence, at moderate Weber number regimes. By analyzing the interfacial strain rate and turbulent kinetic energy exchange, we provide new insights into the interplay between turbulence and bubble breakup dynamics. Different bubble sizes around the integral scale are simulated for a fixed bubble Weber number, which defines the ratio of turbulent and surface tension forces, maintaining the level of deformability. Results reveal that bubbles closest to the turbulence integral scale have the highest deformation levels and are most efficient in extracting energy from the flow. Smaller bubbles show the least amount of deformation and are unable to capture energy from the larger eddies, in agreement with the Kolmogorov-Hinze hypothesis.
... The advection equation for the level-set is discretized in space using a fifth-order weighted essentially non-oscillatory scheme [23] and advanced in time with a thirdorder total variation diminishing Runge-Kutta scheme [24]. A dual-redistancing approach is used at every time step to ensure the level-set remains a signed distance function while maintaining global mass conservation of each phase [25,26]. Numerical tests show that this dual-redistancing approach works very well and achieves low errors in mass conservation, while avoiding the additional complexities of other approaches such as coupled Volume of Fluid (CVOF) [22]. ...
Two-phase flows involving the dynamic interaction of gas and liquid phases are fundamental to both natural and industrial systems. A critical phenomenon in these flows is bubble fragmentation, which affects interfacial area and mass/momentum transfer. Direct Numerical Simulations (DNS) of turbulent two-phase bubbly flows allow for an improved control of physical parameters and access to local flow variables which are challenging to obtain from traditional experiments. In this work we perform numerical experiments of bubble breakup in decaying isotropic turbulence, at moderate Weber number regimes. By analyzing the interfacial strain rate and turbulent kinetic energy exchange, we provide new insights into the interplay between turbulence and bubble breakup dynamics. Different bubble sizes around the integral scale are simulated for a fixed bubble Weber number, which defines the ratio of turbulent and surface tension forces, maintaining the level of deformability. Results reveal that bubbles closest to the turbulence integral scale have the highest deformation levels, and are most efficient in extracting energy from the flow. Smaller bubbles show the least amount of deformation, and are unable to capture energy from the larger eddies, in agreement with the Kolmogorov-Hinze hypothesis.
... For instance, the outward unit normal vector to Γ(τ ) is given by n = ∇φ |∇φ| , and the mean curvature of each level set is κ = ∇ · ∇φ |∇φ| . Other geometric quantities, such as the arclength |Γ| and the enclosed area |ω| of ω, can be expressed respectively as 26,28 : ...
We propose here the use of the variational level set methodology to capture Lagrangian vortex boundaries in 2D unsteady velocity fields. This method reformulates earlier approaches that seek material vortex boundaries as extremum solutions of variational problems. We demonstrate the performance of this technique for two different variational formulations built upon different notions of coherence. The first formulation uses an energy functional that penalizes the deviation of a closed material line from piecewise uniform stretching [Haller and Beron-Vera, J. Fluid Mech. 731, R4 (2013)]. The second energy function is derived for a graph-based approach to vortex boundary detection [Hadjighasem et al., Phys. Rev. E 93, 063107 (2016)]. Our level-set formulation captures an a priori unknown number of vortices simultaneously at relatively low computational cost. We illustrate the approach by identifying vortices from different coherence principles in several examples.
... and the discretized Heaviside function [38] is ...
We investigate the evolution of vortex-surface fields (VSFs) in compressible Taylor--Green flows at Mach numbers (Ma) ranging from 0.5 to 2.0 using direct numerical simulation. The formulation of VSFs in incompressible flows is extended to compressible flows, and a mass-based renormalization of VSFs is used to facilitate characterizing the evolution of a particular vortex surface. The effects of the Mach number on the VSF evolution are different in three stages. In the early stage, the jumps of the compressive velocity component near shocklets generate sinks to contract surrounding vortex surfaces, which shrink vortex volume and distort vortex surfaces. The subsequent reconnection of vortex surfaces, quantified by the minimal distance between approaching vortex surfaces and the exchange of vorticity fluxes, occurs earlier and has a higher reconnection degree for larger Ma owing to the dilatational dissipation and shocklet-induced reconnection of vortex lines. In the late stage, the positive dissipation rate and negative pressure work accelerate the loss of kinetic energy and suppress vortex twisting with increasing Ma.
... In particular, the isoline at φ s = 1/2 represents the interface. Similarly to the Volume of Fluid (Hirt & Nichols 1981) and Level Set (Sussman et al. 1994;Chang et al. 1996) methods used to simulate multi phase flows, we can write the stress in a mixture form as ...
We perform numerical simulations of a turbulent channel flow over an hyper-elastic wall. In the fluid region the flow is governed by the incompressible Navier-Stokes (NS) equations, while the solid is a neo-Hookean material satisfying the incompressible Mooney-Rivlin law. The multiphase flow is solved with a one-continuum formulation, using a monolithic velocity field for both the fluid and solid phase, which allows the use of a fully Eulerian formulation. The simulations are carried out at Reynolds bulk Re=2800 and examine the effect of different elasticity and viscosity of the deformable wall. We show that the skin friction increases monotonically with the material elastic modulus. The turbulent flow in the channel is affected by the moving wall even at low values of elasticity since non-zero fluctuations of vertical velocity at the interface influence the flow dynamics. The near-wall streaks and the associated quasi-streamwise vortices are strongly reduced near a highly elastic wall while the flow becomes more correlated in the spanwise direction, similarly to what happens for flows over rough and porous walls. As a consequence, the mean velocity profile in wall units is shifted downwards when shown in logarithmic scale, and the slope of the inertial range increases in comparison to that for the flow over a rigid wall. We propose a correlation between the downward shift of the inertial range, its slope and the wall-normal velocity fluctuations at the wall, extending results for the flow over rough walls. We finally show that the interface deformation is determined by the fluid fluctuations when the viscosity of the elastic layer is low, while when this is high the deformation is limited by the solid properties.
... However, it can potentially involve drawbacks with respect to accuracy of mass conservation, resolution of both discontinuities and high gradients across the interface, or the imposition of boundary conditions. The most common examples for interface capturing methods are the level-set method [2,3] and the volume-of-fluid method [4]. ...
A novel method - the Virtual Ring Shear-Slip Mesh Update Method (VR-SSMUM) - for the efficient and accurate modeling of moving boundary or interface problems in the context of the numerical analysis of fluid flow is presented. We focus on cases with periodic straight-line translation including object entry and exit. The periodic character of the motion is reflected in the method via a mapping of the physical domain onto a closed virtual ring. Therefore, we use an extended mesh, where unneeded portions are deactivated to control the computational overhead. We provide a validation case as well as examples for the applicability of the method to 2D and 3D models of packaging machines.
... This is obtained by introducing a monolithic velocity vector field valid everywhere, found by applying the volume averaging procedure [35,36]. Thus, we can write the Cauchy stress tensor σ ij in a mixture form, similarly to the Volume of Fluid [37] and Level Set [38,39] methods commonly used to simulate multiphase flows: ...
We consider suspensions of deformable particles in a Newtonian fluid by means of fully Eulerian numerical simulations with a one-continuum formulation. We study the rheology of the visco-elastic suspension in plane Couette flow in the limit of vanishing inertia and examine the dependency of the effective viscosity on the solid volume-fraction , the capillary number \mbox{Ca}, and the solid to fluid viscosity ratio \mbox{K}. The suspension viscosity decreases with deformation and applied shear (shear-thinning) while still increasing with volume fraction. We show that collapses to an universal function, , with an effective volume fraction , lower than the nominal one owing to the particle deformation. This universal function is well described by the Eilers fit, which well approximate the rheology of suspension of rigid spheres at all . We provide a closure for the effective volume fraction as function of volume fraction and capillary number \mbox{Ca} and demonstrate it also applies to data in literature for suspensions of capsules and red-blood cells. In addition, we show that the normal stress differences exhibit a non-linear behavior, with a similar trend as in polymer and filament suspensions. The total stress budgets reveals that the particle-induced stress contribution increases with the volume fraction and decreases with deformability.
... The level set (LS) method, originally developed for tracking moving boundaries [1][2][3][4] has been widely applied in various TO problems [5][6][7][8][9]. TO problems can be cast into LS update equations using the shape sensitivities of corresponding objective functions and constraints [7,9]. ...
This work presents a comparative study on the application of isogeometric analysis in boundary variation methods based topology optimization problems. Level set and phase field are two boundary variation methods gaining in popularity in topology optimization research community. Among different formulations on update methods, this work employs reaction-diffusion and Allen-Cahn update equations for level set and phase field methods respectively. The application of the isogeometric analysis method for these two update equations is new in the literature. The work explores the effect of different parameters, like basis function order, diffusion coefficient, and mesh size on three benchmark topology optimization problems. Our results indicate that quadratic NURBS-basis functions are adequate to solve compliance minimization problems, and increasing order leads to an increase of computation time without much change in accuracy. We found that whereas the phase field method allows a range of diffusion coefficients, the reaction-diffusion-based level set method was able to converge only for a narrow range of diffusion coefficients. Both methods faced convergence problems when the mesh size is increased for higher order basis functions. This study can provide guidance to new users interested in the application of isogeometric analysis in boundary variation methods for topology optimization.
... The motion of curves or surfaces with normal velocity that depends on curvature has a wide range of applications in science, engineering, and mathematics. A short, and nowhere near complete list includes materials science [1,2], fluid and bubble motion [3,4], image processing [5], computer vision [6,7], stochastic control [8], and more recently, data science [9]. There is a wealth of literature on numerical schemes for approximating geometric motions, and one of the most successful and widely used algorithms is the level set method. ...
We introduce a novel algorithm that converges to level set convex viscosity solutions of high-dimensional Hamilton–Jacobi equations. The algorithm is applicable to a broad class of curvature motion PDEs, as well as a recently developed Hamilton–Jacobi equation for the Tukey depth, which is a statistical depth measure of data points. A main contribution of our work is a new monotone scheme for approximating the direction of the gradient, which allows for monotone discretizations of pure partial derivatives in the direction of, and orthogonal to, the gradient. We provide a convergence analysis of the algorithm on both regular Cartesian grids and unstructured point clouds in any dimension, and present numerical experiments that demonstrate the effectiveness of the algorithm in approximating solutions of the affine flow in two dimensions and the Tukey depth measure of high-dimensional datasets such as MNIST and FashionMNIST.
... On the other hand, interface-capturing methods, such as Volume-of-Fluid (VOF) [9] and level set methods [10], do not require the mesh to conform to the interface. Instead, these methods implicitly represent the interface using a scalar field. ...
Traditional methods for solving physical equations in curved spaces, especially in fluid mechanics and general relativity, rely heavily on the use of Christoffel symbols. These symbols provide the necessary corrections to account for curvature in differential geometries but lead to significant computational complexity, particularly in numerical simulations. In this paper, we propose a novel, simplified approach that obviates the need for Christoffel symbols by symbolic programming and advanced numerical methods. Our approach is based on defining a symbolic mapping between Euclidean space and curved coordinate systems, enabling the transformation of spatial and temporal derivatives through Jacobians and their inverses. This eliminates the necessity of using Christoffel symbols for defining local bases and tensors, allowing for the direct application of physical laws in Cartesian coordinates even when solving problems in curved spaces. We demonstrate the robustness and flexibility of our method through several examples, including the derivation of the Navier-Stokes equations in cylindrical coordinates, the modeling of complex flows in bent cylindrical tubes, and the breakup of viscoelastic fluid threads. These examples highlight how our method simplifies the numerical formulation while maintaining accuracy and efficiency. Additionally, we explore how these advancements benefit free-surface flows, where mapping physical 3D domains to a simpler computational domain is essential for solving moving boundary problems.
... A dual-redistancing approach is used at every time step to ensure the level-set remains a signed distance function while maintaining global mass conservation of each phase Chang et al. (1996); Zhang et al. (1998). Numerical tests show that this dual-redistancing approach works very well and achieves low errors in mass conservation, while avoiding the additional complexities of other approaches such as coupled Volume of Fluid (CVOF) Calado et al. (2024). ...
A ship wake has its characteristic signature from breaking waves at the bow, a turbulent wake, and "white water" surrounding it due to entrained air diffracting light beams. The flow physics are complex, involving gas and liquid phases, turbulence and multi-scale phenomena. High-fidelity computations that capture the interface require intensive resources, particularly in the high Reynolds (turbulent) regime. Direct Numerical Simulations of quasi-steady forced turbulence below a free-surface are conducted as a model for the air entrainment behind a ship's wake. The two-phase flow solver uses the fractional step method for the solution of the Navier-Stokes equation, with a conservative level-set approach to capture the interface, and fast Poisson equation solution. Statistically stationary turbulence is forced below the free-surface, leading to a dynamic interaction and deformation of the interface and bubble formation. Phase-averaged statistics at iso-depths are performed to analyze air volume fraction, turbulent length scales and Reynolds stress components. The Reynolds stresses show a transition from isotropy toward quasi-2D turbulence approaching the free-surface. Spectral information on the fluctuating velocities is also extracted within the intermittent free-surface region and compared for different Weber numbers, as well as interpolated quantities from the free-surface itself. Statistics on the total interfacial area, bubble sizes and sphericity are also analyzed.
... The advection equation for the level-set is discretized in space using a fifth-order weighted essentially non-oscillatory (WENO-5) scheme 23 and advanced in time with a third-order total variation diminishing (TVD) Runge-Kutta scheme 24 . A dualredistancing approach is used at every time step to ensure the level-set remains a signed distance function while maintaining global mass conservation of each phase 25,26 . Numerical tests show that this dual-redistancing approach achieves low errors in mass conservation, while avoiding the additional complexities of other approaches such as coupled Volume of Fluid (CVOF) 22 . ...
Bubble deformation and breakup from turbulence is present in many engineering applications and in nature, yet the physical mechanisms still remain poorly understood. Depending on the local turbulence intensity or Weber number, a bubble may either deform without breakup, suffer a violent breakup, or exhibit a resonant behavior, where the turbulent eddies excite the bubble's natural frequencies. Recent studies have used spherical harmonic decomposition to analyze bubble interaction with turbulence, quantifying the deformation energy of each eigenmode. However, this approach is only applicable for small levels of deformation (linear regime), while the bubble shape remains close to a sphere. In the present work, we present a novel data-driven approach combining large deformation diffeomorphic metric mapping and proper orthogonal decomposition, which is more robust for large deformations. The method is tested on a set of validation cases and applied to turbulent bubble deformation cases obtained from direct numerical simulations data.
... Following Chang et al., 57 to regularize the viscosity, the Heaviside function H is substituted by the subsequent smoothed version as follows: ...
We investigated the effects of bubble count, flow direction, and Eötvös number on deformable bubbles in turbulent channel flow. For a given shear Reynolds number Re = 180 and fixed bubble volume fractions (1.263% and 2.525%), we conducted a series of direct numerical simulations using a coupled level-set and volume-of-fluid solver to evaluate their impact on bubble volume fraction distribution, velocity fields, and turbulence characteristics. Each aspect was studied based on the microscopic equations of two-phase flow, and the accuracy of the modeling terms used in current Reynolds-averaged Navier–Stokes equation (RANS) models was assessed. The influence on the anisotropic state was analyzed using the Lumley triangle, and the anisotropy of Reynolds stresses was captured through the exact balance equations. The results indicate that in upward flow, bubbles tend to accumulate near the wall, with smaller Eötvös numbers leading to closer proximity to the wall and greater attenuation of the liquid-phase velocity. This distribution enhances energy dissipation and turbulence isotropy. In downward flow, bubbles cluster in the channel center, generating additional pseudo-turbulence and attenuating energy in the buffer layer. Moreover, the interfacial transfer of turbulent energy, as currently modeled in RANS, is found to be inadequate for upward flows.
... Following Chang et al. [40], to regularize the viscosity, the Heaviside function H is substituted by the subsequent smoothed version as: ...
This study explores the dynamics of dispersed bubbly turbulent flow in a channel using interface-resolved direct numerical simulation (DNS) with an efficient Coupled Level-Set Volume-of-Fluid (CLSVOF) solver. The influence of number of bubbles (96 and 192), flow direction, and Eotvos number was examined across eight distinct cases. The results indicate that in upward flows, bubbles tend to accumulate near the wall, with smaller Eotvos numbers bringing them closer to the wall and enhancing energy dissipation through increased turbulence and vorticity. This proximity causes the liquid phase velocity to attenuate, and the bubbles, being more spherical, induce more isotropic turbulence. Conversely, in downward flows, bubbles cluster in the middle of the channel and induce additional pseudo-turbulence in the channel center, which induce additional turbulent kinetic energy in the channel center. The study further examines budget of Turbulent Kinetic Energy (TKE) and the exact balance equation for the Reynolds stresses, revealing that near-wall bubble motion generates substantial velocity gradients, particularly in the wall-normal direction, significantly impacting the turbulence structure.
... For this reason, these forces and the resulting interface evolution must be calculated accurately. The numerical procedures devised to deal with the interface evolution are typically classified into interface-capturing [ 9,10] and interface-tracking [ 11] methods. ...
This chapter briefly describes the theoretical approaches commonly used to gain insight into the tip streaming phenomenon. These approaches allow one to solve or at least obtain information from the governing equations presented in the previous chapter.
We discuss the main characteristics of direct numerical simulations and global and local stability analyses. The differences between these two last approaches are explained to emphasize the importance of the global stability analysis for tip streaming. Concepts such as global modes, asymptotic stability, and short-term response are also discussed.
The temporal and spatiotemporal local stability analyses provide information on the behavior of the jets emitted in the microjetting mode of the tip streaming configurations. This chapter closes with some results obtained from those analyses and directly related to those configurations, including the effects of electric fields, surfactants, and viscoelasticity.
... Stokes interface problems arise from the simulation of the two-phase incompressible flows (cf. [11,29,32]), fluid structure interaction (FSI) problems (cf. [24,30,32]), and other applications whose simulation domain is often formed by several materials separated by curves or surfaces from each other. ...
In this paper, we present a high-order weak Galerkin finite element method (WG-FEM) for solving the stationary Stokes interface problems with discontinuous velocity and pressure in ℝ d ( d = 2, 3). This WG method is equipped with stable finite elements consisting of usual polynomials of degree k ≥ 1 for the velocity and polynomials of degree k – 1 for the pressure, both are discontinuous. Optimal convergence rates of order k + 1 for the velocity and order k for the pressure are established in L ² -norm on hybrid meshes. Numerical experiments verify the expected order of accuracy for both two-dimensional and three-dimensional examples. Moreover, numerically it is shown that the proposed WG algorithm is able to accommodate geometrically complicated and very irregular interfaces having sharp edges, cusps, and tips.
... where ∆t is the time-step and the D v = 1 2 (∇U + (∇U ) T ) is the deformation tensor. The convective terms in the momentum equation are discretized using the higher order essentially non-oscillatory (ENO) scheme as described by Chang et al. (1996), and the remaining space derivatives are discretized using the central difference scheme. The discretized form of the momentum equation is solved explicitly for the known volume fraction field F n , which gives rise to the provisional velocity field. ...
The dynamics of two vertically coalescing drops and a pool of the same liquid have been investigated using a Coupled Level Set and Volume of Fluid (CLSVOF) method. Such a configuration enables us to study the dynamic interaction of an arbitrary-shaped liquid conglomerate, formed owing to drop-drop coalescence, with a pool. Similar to drop-pool and drop-drop interactions, partial coalescence is observed when a conglomerate interacts with a pool. The presence of the pool below the father drop is found to influence the coalescence characteristic of the two drops. At the same time, the movement of the capillary waves resulting from the interaction of two drops governs the coalescence dynamics of the conglomerate with the pool. As liquid interfaces interact and generate capillary waves at multiple locations, complex trajectories of capillary waves are observed, which play a crucial role in determining the pinch-off characteristics of the satellite during conglomerate-pool interaction. We examine the effect of the ratio of the diameters of the lower/father drop to the upper/mother drop (D_r) on the coalescence dynamics while maintaining the size of the mother drop constant. The variation in the coalescence dynamics due to change in is quantified in terms of the residence time (tau_r), pinch-off time (tau_p) and the satellite diameter to conglomerate diameter ratio (Ds/Dc). The coalescence dynamics of the conglomerate is then compared with that of an equivalent spherical drop of the same volume and also with that of a drop initialized with the same shape as that of the conglomerate. Finally, the regions of complete and partial coalescence for the conglomerate-pool interactions are demarcated on the Weber number - diameter ratio (We-Dr) space.
Pool boiling occurs in numerous applications, including the space industry, nuclear reactors, and thermal packaging. Liquid subcooling is defined as the difference between the saturation temperature and the bulk liquid temperature, which is known to affect heat transfer in certain thermal regimes because of its impact on the formation, growth, and dynamics of vapor bubbles. However, the precise nature and magnitude of its effects on the primary characteristics of nucleate boiling are currently not fully understood. Moreover, the available data in the literature currently indicate divergent trends. This work utilizes phase-resolving direct numerical simulations to elucidate the effects of subcooling in pool boiling in an attempt to shed light on the current uncertainty in the literature. We conducted a series of simulations where the bulk liquid temperature was varied, while the wall temperature and saturation temperature were kept constant. Bubble formation and departure, as well as their properties, such as equivalent bubble diameter, centroid coordinates, surface area, and bubble volume, were estimated for all computations. Our results demonstrate that the bulk liquid temperature affects the heat flux curve as well as bubble dynamics, while the latter exhibits a non-monotonic behavior.
The principle of electrohydrodynamics (EHD) processes relies on manipulating fluids using electric forces. The advantage of EHD over other fluid manipulation approaches, such as thermal or acoustic-based processes, consists in its excellent controllability, versatility (in terms of the range of suitable fluids), and relatively low cost. The importance of modeling and simulation of EHD processes, particularly for the microsystems, has been growing over the past decade, replacing on many occasions trial-and-error approaches. The present paper is devoted to the advances in the numerical modeling and simulation of electrohydrodynamic (EHD) problems. Physical phenomena playing an essential role in EHD are explained and governing equations are formulated. Major challenges faced when modeling EHD problems are highlighted and different classes of numerical approaches used for handling them are outlined. Benefits and disadvantages as well as open issues in different numerical approaches are also discussed. Finally, the paper provides an overview of numerical studies of EHD in multi-phase micro-systems emphasizing some key findings for three classes of problems, namely droplets, jets and planar films exposed to external electric fields.
We consider the inverse problem of finding guiding pattern shapes that result in desired self-assembly morphologies of block copolymer melts. Specifically, we model polymer self-assembly using the self-consistent field theory and derive, in a non-parametric setting, the sensitivity of the dissimilarity between the desired and the actual morphologies to arbitrary perturbations in the guiding pattern shape. The sensitivity is then used for the optimization of the confining pattern shapes such that the dissimilarity between the desired and the actual morphologies is minimized. The efficiency and robustness of the proposed gradient-based algorithm are demonstrated in a number of examples related to templating vertical interconnect accesses (VIA).
In this paper, the authors propose a new model for active contours segmentation in a given image, based on Mumford-Shah functional (Mumford and Shah, 1989). The model is composed of a system of differential and integral equations. By the experimental results we can keep the advantages of Chan and Vese's model (Chan and Vese, 2001) and avoid the regularization for, Dirac function. More importantly, in theory we prove that the system has a unique viscosity solution.
Liquid nitrogen is a widely used cryogenic fluid in various industrial and scientific applications. Film boiling is inevitable when a hot surface is exposed to a cryogenic fluid like liquid nitrogen. This mode of boiling can affect the heat transfer rate and cooling efficiency of the process. The present study investigated the effect of operating pressure on film boiling characteristics of liquid nitrogen using continuum-based numerical simulations. Simulations were performed at atmospheric, intermediate, and near-critical pressures using a coupled level-set and volume of fluid method of interface tracking. Bubble dynamics and heat flux variations were examined at different degrees of wall superheat. The results show that the pressure condition significantly affects the bubble dynamics and boiling of liquid nitrogen. At higher pressure, the specific heat of the vapor phase increases, resulting in a higher heat transfer rate. It was also noted that at high levels of superheat, the bubbles detach from the heating surface faster, resulting in a higher transfer of heat. The effect of pressure on the number of bubble formation sites was also investigated. The simulations also reveal that the local thermodynamic conditions, such as pressure, temperature, and vapor concentration, significantly influence the bubble size.
Microalgae, at present, has become a frontier of the green biotechnology and has been given widespread attention as an ideal feedstock for biofuel production. Generally, sustainable economic growth, social and environmental condition and human activity largely depend on utilization of the energy sources. As developing nation rises up coupled with expansion of population, the demand of fossil fuel energy is now sky-rocketing. However, continuous use of liquid fossil fuel is unsustainable due to depleting sources. So, alternatives to fossil fuel engage scientists and entrepreneurs around the world. Micro algae based biofuel is one of the alternatives to fossil fuel. In microalgae based biofuel system, the main optimal factors for microalgae cultivation are light, CO2, temperature and pH . The light is one of the main factors for photosynthesis and microalgae grow photo-synthetically. The CO2 gas is the main carbon source for photosynthetic culture of microalgae .The optimal growth of microalgae depends on CO2 requirement with different species of algae and geometries of Photobioreactor (PBR). During the cultivation process the gas CO2 is injected in the tubular PBR. But CO2 creates bubbles inside the PBR which impacts flow patterns of the microalgae suspension and consequently microalgae production is affected. This fact should be considered in case of production processes. This study aims to investigate the effect of CO2 on microalgae growth culture for a tubular phatobioreactor at the geographical location of Chittagong University of Engineering & Technology (CUET). A 20.94m long and 0.025m tubular PBR is considered for the simulation. To observe the microalgae cell growth, we selected the 21st June for a bright sunny and the longest day of a year. From the simulation after day seven we observed a very slow growth for the microalgae culture.It is noted that the growth related to concentration of microalgae is increased by day length with respect to continuous sunlight. A small fluctuation of shear rate around U loop area is also found in our simulation which may be caused to reduce the volumetric production due to cell fragility. From the velocity profile we found that, the velocity is generally higher in the middle of the tube which gives a parabolic shape of the suspension flow.
In the context of simulation‐based methods, multiple challenges arise, two of which are considered in this work. As a first challenge, problems including time‐dependent phenomena with complex domain deformations, potentially even with changes in the domain topology, need to be tackled appropriately. The second challenge arises when computational resources and the time for evaluating the model become critical in so‐called many query scenarios for parametric problems. For example, these problems occur in optimization, uncertainty quantification (UQ), or automatic control, and using highly resolved full‐order models (FOMs) may become impractical. To address both types of complexity, we present a novel projection‐based model order reduction (MOR) approach for deforming domain problems that takes advantage of the time‐continuous space‐time formulation. We apply it to two examples that are relevant to engineering or biomedical applications and conduct an error and performance analysis. In both cases, we are able to drastically reduce the computational expense for a model evaluation and, at the same time, maintain an adequate accuracy level, each compared to the original time‐continuous space‐time full‐order model (FOM). All in all, this work indicates the effectiveness of the presented MOR approach for deforming domain problems by taking advantage of a time‐continuous space‐time setting.
The evaporation of a liquid drop of initial diameter (Ddrop) migrating in a tube of diameter (D0) is investigated using the coupled level set and volume of fluid (CLSVOF) method focusing on determining the heat and mass transfer coefficients for a deforming drop. A robust phase change model is developed using an embedded boundary method under a finite difference framework to handle vaporizing flows. The model is extensively validated through simulations of benchmark problems such as arbitrary evaporation of a static drop and reproduction of psychrometric data. The results show that the Sherwood number (Sh) and the Nusselt number (Nu) reach a steady value after an initial transient period for the drop subjected to Hagen-Poiseuille flow. A parametric study is conducted to investigate the effect of drop deformation on the rate of evaporation. It is observed that Stefan flow due to evaporation has a negligible impact on the drop deformation dynamics. We also observed that, for different values of Ddrop/D0, the Sh follows a linear correlation with Re1/2Sc1/3.
Compelling evidence shows the association of inflammation with atherosclerosis diseases, one of the leading cause of mortality and morbidity worldwide. Recent research indicated that the inflammatory process of atherosclerotic lesions is involved in the progression of atherosclerotic plaques in specific regions, such as the carotid bifurcation, which represents a risk for ischemic stroke as a result of the interaction between the blood and the plaque. We start modeling using 3D idealized geometry in order to capture the most important features of such interactions. Then, we proceed to a partly patient-specific computational domain representing an atherosclerotic artery. Understanding such interactions is of paramount importance preventing the risk of the plaque rupture. The numerical results comparisons have shown that, qualitatively, there is an agreement between idealized atherosclerotic artery and patient-specific atherosclerotic carotid artery. The idealized carotid geometry will be useful in future FSI studies of hemodynamic indicators based on medical images.
The equations governing periodic vortex sheet roll-up from analytic initial data are desingularized. Linear stability analysis shows that this diminishes the vortex sheet model's short wavelength instability, yielding a numerically more tractable set of equations. Computational evidence is presented which indicates that this approximation converges, beyond the critical time of singularity formation in the vortex sheet, if the mesh is refined and the smoothing parameter is reduced in the proper order. The results suggest that the vortex sheet rolls up into a double branched spiral past the critical time. It is demonstrated that either higher machine precision or a spectral filter can be used to maintain computational accuracy as the smoothing parameter is decreased. Some conjectures on the model's long time asymptotic state are given.
A new method for modeling surface tension effects on fluid motion has been developed. Interfaces between fluids of different properties, or "colors", are represented as transition regions of finite thickness, across which the color variable varies continuously. At each point in the transition region, a force density is defined which is proportional to the curvature of the surface of constant color at that point. It is normalized so that the conventional description of surface tension on an interface is recovered when the ratio of local transition region thickness to local radius of curvature approaches zero. The continuum method eliminates the need for interface reconstruction, simplifies the calculation of surface tension, enables accurate modeling of two- and three-dimensional fluid flows driven by surface forces, and does not impose any modeling restrictions on the number, complexity, or dynamic evolution of fluid interfaces having surface tension. Computational results for two-dimensional flows are given to illustrate the properties of the method.
A coupled level set method for the motion of multiple junctions is proposed. The new method extends the "Hamilton-Jacobi" level set formulation of Osher and Sethian. It retains the feature of tracking fronts by following level sets and allows the specification of arbitrary velocities on each front. The diffusion equation is shown to generate curvature dependent motion and this is used to develop an algorithm to move multiple junctions with curvature-dependent speed. Systems of reaction diffusion equations are shown to possess inherent properties which prohibit efficient numerical solutions when applied to curvature-dependent motion.
In this paper we describe finite element computations of the free-surface flow of a viscous fluid down an undulating inclined plane. The technique developed here employs an orthogonal mapping that is computed along with the velocity and pressure. This is allied to a technique to compute symbolically the Jacobian and other derivatives required for numerical continuation methods. The solutions obtained are compared with laboratory experiments and finite element computations reported by Pritchard and co-workers. The finite element computational method used by these authors employs spines to represent the free surface. An excellent agreement is shown to exist between the new computations and the laboratory experiments, and with the numerical solutions of Pritchard and co-workers.
An inviscid boundary-integral method is modified in order to study the single-scale Rayleigh-Taylor instability for arbitrary Atwood number. The primary modification uses vortex blobs to smooth the Green's function and suppress a finite time singularity in the curvature. Additional modifications to earlier codes such as using second-order central differences along the interface to accommodate spikes in the vorticity and spreading the nodes evenly along the interface to suppress clustering of nodes are designed to maintain resolution and accuracy. To achieve second-order accuracy in time when the nodes are spread, an extra predictor step is needed that shifts the nodes before the variables are advanced. The method successfully follows the development of a single mode to states with asymptotic velocities for the bubble and spike that depend on the Atwood number and are independent of the blob size. Incipient droplet formation is observed.C 1988 Academic Press, Inc.
This paper describes a second-order projection method for variable-density incompressible flows. The method is suitable for both finite amplitude density variations and for fluids that are modeled using a Boussinesq approximation. It is based on a second-order fractional step scheme in which diffusion-convection terms are advanced without enforcing the incompressibility condition and the resulting intermediate velocity field is then projected onto the space of discretely divergence-free vector fields. The nonlinear convection terms are treated using a Godunov-type procedure that is second order for smooth flow and remain stable and non-oscillatory for nonsmooth flows with low fluid viscosities. The method is described for finite-amplitude density variation and the simplifications for a Boussinesq approximation are sketched. Numerical results are presented that validate the convergence properties of the method and demonstrate its performance on more realistic problems.
A method to simulate unsteady multi-fluid flows in which a sharp interface or a front separates incompressible fluids of different density and viscosity is described. The flow field is discretized by a conservative finite difference approximation on a stationary grid, and the interface is explicitly represented by a separate, unstructured grid that moves through the stationary grid. Since the interface deforms continuously, it is necessary to restructure its grid as the calculations proceed. In addition to keeping the density and viscosity stratification sharp, the tracked interface provides a natural way to include surface tension effects. Both two- and three-dimensional, full numerical simulations of bubble motion are presented.
We devise new numerical algorithms, called PSC algorithms, for following fronts propagating with curvature-dependent speed. The speed may be an arbitrary function of curvature, and the front also can be passively advected by an underlying flow. These algorithms approximate the equations of motion, which resemble Hamilton-Jacobi equations with parabolic right-hand sides, by using techniques from hyperbolic conservation laws. Non-oscillatory schemes of various orders of accuracy are used to solve the equations, providing methods that accurately capture the formation of sharp gradients and cusps in the moving fronts. The algorithms handle topological merging and breaking naturally, work in any number of space dimensions, and do not require that the moving surface be written as a function. The methods can be also used for more general Hamilton-Jacobi-type problems. We demonstrate our algorithms by computing the solution to a variety of surface motion problems.
Two Lagrangian-Eulerian vortex methods to simulate the motion of an interface between inviscid fluids of different densities are presented. The representation of the interface as a vortex sheet eliminates numerical diffusion, and by coupling the tracked interface with a stationary grid (using the well-known vortex-in-cell method) the high cost associated with traditional vortex methods is reduced. These methods are applied to the Rayleigh-Taylor instability. For finite density ratios the appearance of rolled up vortices and a possible singularity formation has limited simulations in the past. By providing proper regularization our methods overcome some of these difficulties.
In this paper we extend our earlier work on the efficient implementation of ENO (essentially non-oscillatory) shock-capturing schemes. We provide a new simplified expression for the ENO construction procedure based again on numerical fluxes rather than cell-averages. We also consider two improvements which we label ENO-LLF (local Lax-Friedrichs) and ENO-Roe, which yield sharper shock transitions, improved overall accuracy, for lower computational cost than previous implementation of the ENO schemes. Two methods of sharpening contact discontinuities—the subcell resolution idea of Harten and the artificial compression idea of Yang, which those authors originally used in the cell average framework—are applied to the current ENO schemes using numerical fluxes and TVD Runge-Kutta time discretizations. The implementation for nonlinear systems and multi-dimensions is given. Finally, many numerical examples, including a compressible shock turbulence interaction flow calculation, are presented.
Nonlinear interfacial gravity waves in a two-layer Boussinesq fluid are studied. In a previous paper, nonlinear waves on a vortex sheet separating two layers, each of constant density and velocity were considered. In the present paper the basic flow model consists of a constant vorticity upper layer bounded by a rigid surface and an irrotational lower layer of infinite depth with continuity of the unperturbed velocity at the density interface. Numerical solutions obtained from an exact formulation in terms of a complex-valued integral equation for the shape and local vortex-sheet strength of the wave profile are compared with results from a second-order Stokes expansion. It is found that the wave of maximum amplitude displays different geometrical features depending on the unperturbed flow parameters. These include waves containing an S-shaped section, waves with cusped crests and surface-constrained waves with long flat crests. Wave integral properties calculated, including the flux of momentum and energy in the wave propagation direction, showed monotonic variation with increasing wave amplitude.