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Local geometry of a weak normal shock wave interacting with turbulence
Physics of Fluids
Abstract
The shock surface geometry is investigated with direct numerical simulations of a weak normal shock wave propagating in turbulence. The geometry is quantified with the principal curvatures of the surface. A large part of the surface has an approximately flat saddle shape, while elliptic concave and convex shapes with a large curvature intermittently appear on the shock surface. The pressure–dilatation correlation in the governing equation of pressure is investigated at the shock wave with the decomposition into three terms associated with the velocity gradients in the two directions of the principal curvatures and the normal direction of the shock wave. Fluid expansion in the tangential direction occurs at the shock wave with a convex shape in the direction of the shock propagation, resulting in a smaller pressure jump across the shock wave. For a concave shape, compression in the tangential direction can amplify the pressure jump. Consistently, small and large shock Mach numbers are observed for convex and concave shapes, respectively. The geometric influences are the most significant for elliptic concave and convex shapes with approximately equal curvatures in the two principal directions because the compression or expansion occurs in all tangential directions. These relations between the shock surface geometry and shock Mach number observed in turbulence are consistent with the theory of deformed shock waves, suggesting that the three-dimensional geometrical features of the shock surface are important in the modulation of shock waves due to turbulence.
... Recent studies are mainly based on experiments [12][13][14][15][16][17][18] and numerical simulations. [19][20][21][22][23][24][25][26][27][28] These studies have revealed changes in the characteristics of shock waves and turbulence resulting from their interaction. In particular, the changes in turbulence characteristics in interaction with shock waves have been extensive. ...
... Indeed, we observed that overpressure due to the shock wave was lower than average when it was ahead of its mean position and higher when it was behind. Another investigation of shock wave deformation was conducted by Kusuhata et al. 28 using the DNS data obtained by our group. 27 They demonstrated a relationship between the local shape of the shock-wave front and local shock Mach numbers using mean and Gaussian curvatures. ...
... The time-evolved turbulence becomes statistically steady by using external linear forcing. 27,28 The size of the computational domain is 8L 0 in each direction, and the number of grid points is 256 3 . The energy spectrum shown by Pope 32 was given as the amplitude, and the phase was given based on random numbers. ...
In this study, we have performed direct numerical simulations (DNSs) of the interaction between a spherical shock wave and homogeneous isotropic turbulence to investigate shock wave modulation. We considered two parameters, the central pressure of the initial high-energy region that generates the shock wave and the turbulent Mach number Mt. DNSs were performed under five conditions. A polar coordinate system (r,θ,ϕ) was then defined for the analyses, with the center of the high-energy region as the origin. The local position of the shock wave rs was defined as the position of the maximum value PM of the radial pressure distribution P(r) at two declinations (θ,ϕ) in the polar coordinate system. Specifically, rs and PM were functions of (θ,ϕ), and their statistics were computed using data over all (θ,ϕ) at each time. The standard deviation of the fluctuation of rs increased monotonically with the shock-wave front propagation in turbulence. This shows that the shock-wave front deformation grew monotonically. The mean pressure distribution conditioned by the fluctuations of rs and joint probability density function of fluctuation of rs and PM show that there is a negative correlation between the deformation of the shock-wave front and the local intensity of the shock wave. This indicates that the deformed shock-wave front tends to return to its original shape. However, the monotonous growth of the deformation indicates the presence of a counter-effect that allows it to grow.
... Furthermore, the presence of upstream dilatational energy, associated with intrinsic compressibility in the pre-shock turbulence, likewise serves to further augment the TKE amplification, particularly at elevated bulk Mach numbers, as evidenced by the numerical simulations of Grube and Martin 105 and Cuadra et al. 56 The study of non-linear interactions between turbulence and shock waves has also led to the identification of three primary regimes of the shock deformations. 46,64,[107][108][109][110][111] When the upstream perturbations are weak, the shock wave remains coherent, and its position is only locally displaced upstream and downstream with respect to its mean location. If the turbulence perturbations that impinge upon the shock are sufficiently strong, they prove able to locally alter the intensity of the discontinuity and even generate holes within its front. ...
Hypersonic flight involves a variety of complex flow phenomena that directly impact the aerothermodynamic loading of high-speed vehicles. The turbulence encountered during a typical flight trajectory influences and interacts with the shock waves on and around the surface of a vehicle and its propulsion system, affecting both aerodynamic and power plant performance. These interactions can be studied by isolating a turbulent flow convected through a normal shock, commonly referred to as the canonical shock-turbulence interaction (STI) problem. Scale-resolving computational fluid dynamics (CFD) and linear interaction analysis (LIA) have been crucial in studying this problem and formulating scaling laws that explain the observed behavior. In this work, an extensive review of the theoretical (LIA) and numerical (CFD) work on the canonical STI is presented. The majority of the work conducted to date has focused on calorically perfect gases with constant heat capacities. However, in hypersonic flows, chemical and thermal non-equilibrium effects may alter the nature of the interaction. As a result, relevant LIA and CFD studies addressing high-enthalpy phenomena are also succinctly discussed.
Direct numerical simulations are performed for flow past circular cylinders by the lattice Boltzmann method coupled with immersed moving boundary method. By analyzing the flows past a single cylinder at a wide range of cross-flow or in-line oscillation amplitude (0.25≤A/D≤1.5) and frequency (0.5≤fe/f0≤1.5), the results find that the vortex shedding modes inside and outside “lock-in” interval are of significant difference. The vortex shedding mode in the “unlock-in” state is 2S, but C(2S) and P + S shedding modes can be found in the lock-in state. Dynamic mode decomposition is used to analyze characteristic flow features, which shows that mode 1 is the main factor reflecting the flow field structure and mode 2 represents the vortex shedding mode in this work. The vortex shedding modes of flows past a tandem and side-by-side cross-flow double oscillating cylinders are systemically investigated. For tandem double oscillation cylinders, the results of modal decomposition suggest that the shear layer of upstream oscillating cylinder is separated behind the downstream cylinder at a space rate of L/D≤2, but separated behind the upstream cylinder at L/D≥3. Mode 2 at L/D=4 differs from other vortex shedding modes due to the strong inhibition effect by the downstream cylinder on the vortex formation of upstream cylinder. For side-by-side double oscillation cylinders, the wake of two cylinders is a single vortex street at H/D=1, a bistable flow at H/D=2 or 3, a coupled vortex street at H/D=4, and close to a single cylinder at H/D>4. The results of modal decomposition are disordered at H/D=2 due to the interaction between two cylinders and effect of gap flow.
Although existing studies have investigated dominant structures in round jets in developed states, the image of the structures still has been unclear because the observations have often been based on velocity correlation measurements from a limited number of measurement points. In this study we analyzed a full spatiotemporal jet data generated by a large-eddy simulation (LES) to detect the dominant structures of a round jet much more clearly than ever. To save computational resources, we simulated a temporally developing round jet with a jet Reynolds number of 70, 000. Because the temporally developing jet field is homogeneous and periodic in both the streamwise and azimuthal directions, we analyzed the flow field using Koopman decomposition (KD). To analyze the huge numerical data of the jet, the data were reduced using temporal proper orthogonal decomposition (POD), and KD was subsequently performed for the POD modes. In the developed state of the round jet, two layers (inner and outer) are formed. In the inner layer, single-helical structures are formed, and it is confirmed that their streamwise wavelength is in good agreement with those in existing spatially developing round jets (Samie et al., J. Fluid Mech. 916:A2, 2021). In the outer layer, we newly find triple-helical structures oriented in the streamwise direction more than the single-helices. Although the energy contained by the triple-helices is smaller than that by the single-helices, they display important contributions to the mean velocity diffusion. Finally, we demonstrate that only five types of helices may approximate the original flow field excellently.
The relation between shearing motions and the turbulent/non-turbulent interfacial (TNTI) layer is studied with direct numerical simulation of a temporally evolving planar jet. Small-scale shear layers are detected with the triple decomposition of the velocity gradient tensor, which is decomposed into shear, rotation, and elongation tensors. The shear layers are found in the turbulent sublayer more frequently than in the turbulent core region although they hardly appear in the viscous superlayer. The shear layers undergo a biaxial strain with stretching in the shear vorticity direction and compression in the interface normal direction. This compressive strain is related to the non-turbulent fluid, which is relatively advected toward the shear layer. The shear layer thickness in the TNTI layer is well predicted by Burgers vortex layer. The velocity jump of the shear layer is about seven times the Kolmogorov velocity both in the turbulent core region and the TNTI layer. However, the layer thickness normalized by the Kolmogorov scale is about 6 in the turbulent core region and decreases in the TNTI layer, where consequently, the shear Reynolds number becomes small. The shear layers have significant contributions to the enstrophy production in the turbulent sublayer and the viscous enstrophy-diffusion toward the viscous superlayer. The shear layer and the outer edge of the TNTI layer have a curvature radius of about 50 times the Kolmogorov scale. The alignment between the shear layer orientation and the interface normal direction confirms that the shear layers near the interface are mostly parallel to the TNTI layer.
In this study, the characteristics of side-projected planar shock waves propagating through grid turbulence were investigated using a counter-driver shock tube. At the location of visualization of the shock–turbulence interactions, the range of the shock Mach numbers was and the representative values of the turbulent Mach numbers were . The interaction length between the shock wave and grid turbulence was in the range approximately from to 300 times the integral length scale of the turbulence. For the interaction involving the strongest turbulence with , the weakest planar shock wave with was largely deformed and could not be detected on the projected shadowgraph and schlieren images. The density changes on the grayscale in the projected images of the shock waves weakened and expanded multidimensionally. This undetectable profile of the shock wave could indicate that the shock wave locally lost discontinuous property change profile. The shock waves with did not show the undetectable profile in the projected image. In these cases, the projected thickness of the shock wave increased with an increase in the interaction length. The increase in the projected thickness became larger as the turbulent Mach number increased, and the shock Mach number decreased.
Wind tunnel experiments are reported for a spherical shock wave propagating through turbulent wakes of a single cylinder, double cylinders, grid-turbulence, and a laminar flow, whose influences on the shock wave are compared. Overpressure behind the shock wave is measured on a plate while streamwise velocity is measured at the flow point between the measurement plate and the location of the shock wave ejection. Average of peak-overpressure observed upon arrival of the shock wave is decreased by the mean velocity defect of the cylinder wake. Root mean squared (rms) peak-overpressure fluctuation divided by the averaged peak-overpressure is increased by turbulence, and it becomes larger with the rms velocity fluctuation. Correlation coefficients are calculated between fluctuations of peak-overpressure and low-pass filtered fluid velocity. The strong positive correlation is found for the fluid at the location where the shock ray toward the pressure measurement point passes. The length scale of velocity fluctuation with the strong correlation is related to the integral length scale of turbulence. In the double-cylinder wake experiments, the shock wave that has passed one cylinder wake interacts again with another cylinder wake before it reaches the measurement plate. The correlation coefficient for the velocity fluctuation of the first wake is weakened by the second wake, and this influence becomes more important when the rms velocity fluctuation of the second wake is larger.
In this work, we use numerical simulation and linear inviscid theory to study the thermodynamic field generated by the interaction of a shock wave with homogeneous isotropic turbulence. Fluctuations in density, pressure, temperature and entropy can play an important role in shock-induced mixing, combustion and energy transfer processes. Data from shock-captured direct numerical simulations (scDNS) are used to investigate the variation of thermodynamic fluctuations for varying shock strengths, and the results are compared with linear interaction analysis (LIA). The density, pressure and temperature variances attain large values at the shock, followed by, in general, a rapid decay in the downstream flow. The rapid variation behind the shock makes it difficult to compare numerical results with theoretical predictions. A threshold method based on instantaneous shock dilatation is used to overcome this problem, and it gives excellent match between scDNS and LIA. We find cases with non-monotonic variation with Mach number as well as local peaks in density fluctuations behind the shock. These are explained in terms of the contribution of the post-shock acoustic and entropy modes in the LIA solution and their cross-correlation. Budget of the transport equations reveals interesting insight into the physics governing the thermodynamic field behind the shock wave. It is found that the variances are primarily determined by the competing effects of dilatational and dissipation mechanisms. The dominant mechanisms are identified for a range of conditions, and their implication for developing predictive models is highlighted.
Due to the short residence time of air in supersonic combustors, achieving efficient mixing in compressible turbulent reactive flows is crucial for the design of supersonic ramjet (Scramjet) engines. In this respect, improving the understanding of shock-scalar mixing interactions is of fundamental importance for such supersonic combustion applications. In these compressible flows, the interaction between the turbulence and the shock wave is reciprocal, and the coupling between them is very strong. A basic understanding of the physics of such complex interactions has already been obtained through the analysis of relevant simplified flow configurations, including propagation of the shock wave in density-stratified media, shock-wave–mixing-layer interaction, and shock-wave–vortex interaction. Amplification of velocity fluctuations and substantial changes in turbulence characteristic length scales are the most well-known outcomes of shock–turbulence interaction, which may also deeply influence scalar mixing between fuel and oxidizer. The effects of the shock wave on the turbulence have been widely characterized through the use of so-called amplification factors, and similar quantities are introduced herein to characterize the influence of the shock wave on scalar mixing. One of the primary goals of the present study is indeed to extend previous analyses to the case of shock-scalar mixing interaction, which is directly relevant to supersonic combustion applications. It is expected that the shock wave will affect the scalar dissipation rate (SDR) dynamics. Special emphasis is placed on the modification of the so-called turbulence–scalar interaction as a leading-order contribution to the production of mean SDR, i.e., a quantity that defines the mixing rate and efficiency. To the best of the authors’ knowledge, this issue has never been addressed in detail in the literature, and the objective of the present study is to scrutinize this influence. The turbulent mixing of a passive (i.e., chemically inert) scalar in the presence of a shock wave is thus investigated using high-resolution numerical simulations. The starting point of the analysis relies on the transport equations of the variance of the mixture fraction, i.e., a fuel inlet tracer that quantifies the mixing between fuel and oxidizer. The influence of the shock wave is investigated for three distinct values of the shock Mach number M, and the obtained results are compared to reference solutions featuring no shock wave. The computed solutions show that the shock wave significantly modifies the scalar field topology. The larger the value of M, the stronger is the amplification of the alignment of the scalar gradient with the most compressive principal direction of the strain-rate tensor, which signifies the enhancement of scalar mixing with the shock Mach number.
The geometry of turbulent/non-turbulent interfaces (TNTIs) arising from flows with and without mean shear is investigated using direct numerical simulations of turbulent planar jets (PJET) and shear free turbulence (SFT), respectively, with Taylor Reynolds number of about Reλ≈100. In both flows, the TNTI is preferentially aligned with the tangent to the TNTI displaying convex, where the turbulent fluid nearby tends to have a stronger enstrophy, more frequently than concave shapes. The different flow configurations are reflected in different orientations of the TNTI with respect to the flow direction (and its normal). While the interface orientation with respect to the mean flow direction in PJET has an influence on the velocity field near the TNTI and the enstrophy production in the turbulent sublayer, there is no particular discernible dependence on the interface orientation in SFT. Finally, the intense vorticity structures or “worms,” which are possibly associated with “nibbling” entrainment mechanism, “feel” the local geometry of the TNTI, and it is shown that in PJET, a smaller local radius of these structures arises in regions near the TNTI where the local TNTI faces the mean flow direction.
A numerical analysis of the interaction between decaying shear free turbulence and quiescent fluid is performed by means of global statistical budgets of enstrophy, both, at the single-point and two point levels. The single-point enstrophy budget allows us to recognize three physically relevant layers: a bulk turbulent region, an inhomogeneous turbulent layer, and an interfacial layer. Within these layers, enstrophy is produced, transferred, and finally destroyed while leading to a propagation of the turbulent front. These processes do not only depend on the position in the flow field but are also strongly scale dependent. In order to tackle this multi-dimensional behaviour of enstrophy in the space of scales and in physical space, we analyse the spectral enstrophy budget equation. The picture consists of an inviscid spatial cascade of enstrophy from large to small scales parallel to the interface moving towards the interface. At the interface, this phenomenon breaks, leaving place to an anisotropic cascade where large scale structures exhibit only a cascade process normal to the interface thus reducing their thickness while retaining their lengths parallel to the interface. The observed behaviour could be relevant for both the theoretical and the modelling approaches to flow with interacting turbulent/nonturbulent regions. The scale properties of the turbulent propagation mechanisms highlight that the inviscid turbulent transport is a large-scale phenomenon. On the contrary, the viscousdiffusion, commonly associated with small scale mechanisms, highlights a much richer physics involving small lengths, normal to the interface, but at the same time large scales, parallel to the interface.
Recent developments in the physics and modeling of interfacial layers between regions with different turbulent intensities are reviewed. The flow dynamics across these layers governs exchanges of mass, momentum, energy, and scalars (e.g., temperature), which determine the growth, spreading, mixing, and reaction rates in many flows of engineering and natural interest. Results from several analytical and linearized models are reviewed. Particular attention is given to the case of turbulent/nonturbulent interfaces that exist at the edges of jets, wakes, mixing layers, and boundary layers. The geometry, dynamics, and scaling of these interfaces are reviewed, and future lines of research are suggested. The dynamics of passive and active scalars is also discussed, including the effects of stratification, turbulence level, and internal forcing. Finally, the modeling challenges for one-point closures and subgrid-scale models are briefly mentioned.
A set of direct numerical simulations of isotropic turbulence passing through a nominally normal shock wave is presented. Upstream of the shock, the microscale Reynolds number is 40, the mean Mach number is 1.3–6.0, and the turbulence Mach number is 0.16–0.38. It is shown that the Kolmogorov scale decreases during the shock interaction, which implies that the grid resolution needed to resolve the viscous dissipation is finer than that used in previous studies. This leads to some qualitative differences with previous work, e.g., a rapid increase in the streamwise vorticity variance behind the shock and large anisotropy of the postshock Reynolds stresses. The instantaneous structure of the shock/turbulence interaction is examined using averages conditioned on the instantaneous shock strength. For locally strong compressions, the flow is characterized by overcompression, followed by an expansion. At points where the shock is locally weak, the profiles differ qualitatively depending on the strength of the incoming turbulence relative to the strength of the shock, as measured by the turbulence and mean Mach numbers, respectively. In the wrinkled shock regime, these profiles are discontinuous and the shock has a simple topology. In the broken shock regime, the weak interaction profiles are smooth without any discontinuity.
The extraction of dynamically relevant structures from time-resolved flow data has commonly be restricted to numerically generated flow fields. Equivalent structures, however, could not be obtained from experimental measurements, since the commonly used mathematical techniques required the explicit or implicit availability of an underlying model equation. A numerical scheme based on a Krylov subspace method for the extraction of dynamic modes directly from flow fields --- without the need to resort to a model equation --- will be introduced. This technique can be applied equally to numerically generated or experimental data and thus provides a means to decompose time-resolved measurements into dynamically dominant structures. The treatment of subdomains, spatially evolving flows, PIV data and simple flow visualizations will be demonstrated; a connection to the proper orthogonal decomposition (POD) technique, which is a byproduct of the dynamic mode decomposition, will be pointed out.
Homogeneous rapid distortion theory is used to study the response of shear flows and axisymmetric turbulence to rapid one-dimensional compression. In the shear flow problem, both normal and oblique compressions are considered. The response of these anisotropic flows to compression is found to be quite different from that of isotropic turbulence. Upon normal compression, the amplification of the streamwise component of kinetic energy and the total kinetic energy in shear flows is higher than that in isotropic turbulence. Also, normal compression decreases the magnitude of the Reynolds shear stress by amplifying the pressure–strain correlation in the shear stress equation. Obliquity of compression (defined as the angle between the directions of shear and compression) is seen to significantly affect the evolution of the Reynolds stresses. For a range of oblique angles from −60° to 60°, the amplification of streamwise kinetic energy and total kinetic energy decrease with increasing magnitude of the oblique angle. Also, the tendency of the shear stress to decrease in magnitude is diminished upon increasing the oblique angle; for large oblique angles the shear stress amplifies. Upon compression along the axis of axisymmetry, the amplification of the streamwise component of kinetic energy is higher for contracted turbulence than for isotropic turbulence, while the amplification of the total kinetic energy is lower. The above results are interpreted in a more general framework. It is shown that the amplification of the streamwise component of kinetic energy is determined by the initial E11(κ1) (x1 is the direction of compression). Flows with u1 at lower κ1 have a lower effect of pressure during compression and hence, higher amplification of u21. The amplification of the total kinetic energy is determined by the initial fraction of energy along the direction of compression (u21/q2) and the initial E11(κ1). Flows with higher initial u21/q2 and with u1 at lower κ1 have a larger amplification of q2.
Direct numerical simulations at Reynolds numbers ranging from Relambda=30 to 160 show that the thickness deltaomega of the turbulent/nonturbulent (T/NT) interface in planar jets is of the order of the Taylor scale deltaomega~lambda, while in shear free, irrotational/isotropic turbulence is of the order of the Kolmogorov microscale deltaomega~ε. It is shown that deltaomega is equal to the radius of the large vorticity structures (LVSs) in this region, deltaomega~RLVS. Thus, the mean shear and the Reynolds number affect the T/NT interface thickness insofar as they define the radial dimension of the LVS near the T/NT interface.
Linear forcing has been proposed as a useful method for forced isotropic turbulence simulations because it is a physically realistic forcing method with a straightforward implementation in physical-space numerical codes [T. S. Lundgren, ``Linearly forced isotropic turbulence,'' Annual Research Briefs (Center for Turbulence Research, Stanford, CA, 2003), p. 461; C. Rosales and C. Meneveau, ``Linear forcing in numerical simulations of isotropic turbulence: Physical space implementations and convergence properties,'' Phys. Fluids 17, 095106 (2005)]. Here, extensions to the compressible case are discussed. It is shown that, unlike the incompressible case, separate solenoidal and dilatational parts for the forcing term are necessary for controlling the stationary state of the compressible case. In addition, the forcing coefficients can be cast in a form that allows the control of the stationary state values of the total dissipation (and thus the Kolmogorov microscale) and the ratio of dilatational to solenoidal dissipation. Linear full spectrum forcing is also compared to its low wavenumber restriction. Low wavenumber forcing achieves much larger Taylor Reynolds number at the same resolution. Thus, high Reynolds number asymptotics can be more readily probed with low wavenumber forced simulations. Since, in both cases, a solenoidal/dilatational decomposition of the velocity field is required, the simplicity of the full spectrum linear forcing implementation in physical-space numerical codes is lost. Nevertheless, low wavenumber forcing can be implemented without using a full Fourier transform, and so is computationally less demanding.
Linear interaction analysis is a popular tool to analyze shock–turbulence interactions. With assumptions that simplify the complicated two-way interactions, linear analysis presents explicit solutions for post-shock turbulence. The present work extends the analytical expressions to pressure-dilatation. Both the streamwise and transverse components are presented as a function of mean Mach number and streamwise location. The proposed solutions for pressure-dilatation agree well with data from direct numerical simulations when the adopted assumptions in linear analysis are met. The spatial evolutions under different flow conditions are presented, and the asymptotic states observed with increasing Mach number are discussed.
Supernovae - explosions of stars - are a central problem in astrophysics since they encapsulate the entire process of stellar evolution and nucleosynthesis. Rayleigh-Taylor (RT) and Richtmyer-Meshkov (RM) instabilities, developing during the supernova blast, lead to intense interfacial RT/RM mixing of the star's materials and couple astrophysical to atomic scales. This work analyzes some fluid dynamic mathematical aspects of the titanic task of supernova's blast. We handle mathematical challenges of RT/RM dynamics in supernova relevant conditions by directly linking the conservation laws governing RT/RM dynamics to symmetry-based momentum model, by exactly deriving the model parameters in the scale-dependent and scale-invariant regimes, and by exploring the special self-similar class for RT/RM interfacial mixing with variable accelerations. We reveal that RT/RM dynamics is strongly influenced by deterministic (the initial and the flow) conditions in the scale-dependent linear and nonlinear regimes and in the self-similar mixing regime. The theory outcomes are consistent with the observations of supernova remnants, explain the results of the scaled laboratory experiments in high energy density plasmas, and yield the design of future experiments for accurate quantification of RT/RM dynamics in supernova relevant conditions. We find that from fluid dynamic mathematical perspectives supernovae can be regarded as an astrophysical initial value problem. Along with the guidance of what explodes at microscopic scales, supernova remnants encapsulate information on the explosion hydrodynamics and the associated deterministic conditions at macroscopic scales. We urge such effects be considered in interpretations of the observational data.
In practical aerodynamic problems, curved shock/boundary layer interaction (CSBLI) is more frequently encountered than the canonical SBLI. Owing to the topological complexity of the flow field brought about by shock curvature, accurate prediction of the interaction length scale of CSBLI is a challenging task. In this work, streamwise and spanwise curvatures are introduced in turn with the aim of establishing an analytical model for the interaction length scale of CSBLI based on conservation of mass. The validity and universality of the model are verified, which reveal the impact of the shock curvatures on the interaction, acting as the form of non-uniformity. The proposed method can be regarded as providing a foundation for further research on CSBLI, opening new perspectives for the investigation of SBLI flow structures.
The effects of vertical confinement on a turbulent shear layer are investigated with large-eddy simulations of a freely developing shear layer (FSL) and a wall-confined shear layer (WSL) that develops between two horizontal walls. In the case of the WSL, the growth of the shear layer is inhibited by the walls. Once the walls prevent the development of the shear layer, highly anisotropic velocity fluctuations become prominent in the flow. These anisotropic velocity fluctuations are recognized as elongated large-scale structures (ELSS), whose streamwise length is much larger than the length scales in the other directions. Spectral analysis confirms that the turbulent kinetic energy is dominated by the ELSS, whose streamwise length grows continuously. A proper orthogonal decomposition can effectively extract a velocity component associated with the ELSS. The isotropy of the Reynolds stress tensor is changed by the presence of the ELSS. These changes in flow characteristics due to the ELSS are not observed in the FSL, where the shear layer thickness increases continuously. These behaviors of the WSL are consistent with those of stably stratified shear layers (SSSLs), where flow structures similar to ELSS also develop when the vertical flow development is confined by the stable stratification. The vertical confinement by the walls or stable stratification strengthens mean shear effects. The flow behavior at large scales in the WSL and SSSL is consistent with rapid distortion theory for turbulence subject to mean shear, suggesting that the development of ELSS is caused by the mean shear.
Implicit large eddy simulations are performed to show control of shock and boundary layer interaction over a natural laminar flow airfoil by harmonic heat transfer on the suction surface at a free-stream Mach number M∞=0.72 and angle of attack of α=0.38°, for which experimental/flight test results exist. Surface heat flux is added in a time-periodic manner with the exciter strip located at different locations in the vicinity of the shock location at x/c≈0.49 for the flow without any control. Four cases of localized excitation with Gaussian distribution centered at x/c=0.45, 0.48, 0.50, and 0.55 on the airfoil surface are considered, with a width of 10% chord. The effects of unsteady heat flux on shock strength, location, and modification of its structure are demonstrated by instantaneous and time-averaged flow quantities. Detailed vortical and entropy contour plots and numerical Schlieren indicate flow features, such as creation and propagation of Kutta waves and their interactions with the shock wave. Time-averaged load distributions reveal a shift in the location and strength of the shock, with altered lift and drag time histories. The Fourier transforms of lift and drag coefficients help explain the alteration of the strength and structures of the shock-induced unsteadiness. The imposed excitation results in an improvement of aerodynamic efficiency (lift to drag ratio). The numerical simulations follow the algorithm developed in Sengupta et al. [Comput. Fluids 88, 19–37 (2013)].
Decay characteristics of turbulent kinetic energy and enstrophy in grid turbulence have been investigated in the far downstream region ( x / M ∼ 10 3: x is the downstream distance from the grid, M is the mesh size of the grid) through wind tunnel experiments using hot-wire anemometry, with the lowest turbulent Reynolds number R e λ ≈ 5. The non-dimensional dissipation rate C ε increases rapidly toward the final stage of the transition period of decay and the profile agrees well with previous direct numerical simulation [W. D. McComb et al., “Taylor's (1935) dissipation surrogate reinterpreted,” Phys. Fluids 22, 061704 (2010)] and theoretical estimation [D. Lohse, “Crossover from high to low Reynolds number turbulence,” Phys. Rev. Lett. 73, 3223 (1994)] at very low R e λ in decaying and stationary isotropic turbulence. The present result of C ε is an update on the experimental data in grid turbulence toward a very low R e λ, where measurements have been absent. The energy spectrum in the dissipation range at very low R e λ deviates from a universal form observed at high Reynolds numbers. The decay rate of enstrophy is proportional to S + 2 G / R e λ (S is the skewness of the longitudinal velocity derivative and G is the destruction coefficient). It is shown that G and S + 2 G / R e λ increase rapidly with decreasing R e λ at very low R e λ, indicating that the effect of enstrophy destruction is dominant in the final stage of the transition period of decay. The profiles of S + 2 G / R e λ against R e λ is well fitted by a power-law function even in the final stage of the transition period of decay.
In this study, losing the shock wave profiles under interactions with grid turbulence was investigated experimentally and theoretically. We demonstrated that the shock wave contrast on side-view schlieren images gradually decreased to an undetectable level in the experiment. This shock wave ‘vanishment’ occurred at a low shock Mach number with a high turbulent Mach number. With a relatively strong shock wave, the contrast of the shock wave remained detectable although the shock wave profile region was expanded. To understand the shock wave vanishment phenomenon during interaction with turbulence, we established a shock wave vanishment model based on the solution of a one-dimensional interaction between a shock wave and forward induced flow. The criterion of the occurrence of local vanishment of the shock wave was derived as Mt >= (Ms^2-1)/Ms, where Mt is the turbulent Mach number and Ms is the shock Mach number. The proposed shock vanishment model involves the effect of the interaction length and the shock wave recovery characteristics. The derived model explains the vanishment of weak shock wave profile during turbulence with an interaction length of ten times the order of the integral scale of the turbulence, as observed in the experiment.
A scale-by-scale kinetic energy budget is analyzed near the turbulent/nonturbulent interfacial (TNTI) layer with direct numerical simulations (DNSs) of a local turbulent front evolving without mean shear (shear-free turbulence). A local volume average is used to decompose the flow variables into their large-scale and small-scale components near the TNTI layer. The kinetic energy and interscale energy flux from large to small scales of motion are shown to be severely depleted for small scales within the viscous superlayer. The forward interscale energy transfer from large to small scales near the TNTI layer is mostly caused by the velocity gradient in the interface normal direction while the velocity gradient in the tangential direction transfers, on average, the energy from small to large scales. The velocity gradients that cause the forward energy transfer near the TNTI layer are associated with a compressive motion in the interface normal direction and a shearing motion due to the velocity in the tangential direction. The pressure diffusion increases the kinetic energy near the interface except at small scales within the TNTI layer. The averaged pressure diffusion term at the small scales within the TNTI layer has negative values, which are consistent with the presence of small-scale vortices within the TNTI layer. The transports by turbulent diffusion and interaction between large and small scales are negatively correlated even near the TNTI layer, and their effects are locally canceled by each other as also observed in other turbulent flows.
Implicit large eddy simulation (ILES) of passive scalar transfer in compressible turbulence is evaluated for subsonic and supersonic turbulent planar jets. The ILES used in this study relies on fully explicit numerical schemes for spatial and temporal discretization and low‐pass and shock‐capturing filters used as an implicit subgrid‐scale (SGS) model. The ILES results are compared with the direct numerical simulation (DNS) database of the same flows. The ILES results exhibit good agreements with the DNS for first‐ and second‐order statistics of velocity and passive scalar. The scalar transport by turbulent velocity fluctuations is well captured by the ILES. The temporal evolution of the jet strongly depends on the jet Mach number, where a higher Mach number results in the delay of jet development. The Mach number dependence of velocity and passive scalar fields is consistent between the ILES and DNS. The low‐pass filters used as the implicit SGS model contribute to the dissipation of turbulent kinetic energy and scalar variance. Under the present numerical conditions, the filters account for about 50% of the dissipation in a fully developed turbulent jet. The dissipation rate in the ILES, which is the sum of the grid‐scale and SGS dissipation rates, is very close to the dissipation rate in the DNS, and the amount of the SGS dissipation is well controlled by the low‐pass filters. The filters also dump numerical oscillations in the velocity field caused by strong pressure waves outside the supersonic jet at the high Mach number.
Direct numerical simulation is performed for analyzing the interaction between a normal shock wave and turbulence. The shock wave is initially located in a quiescent fluid and propagates into a local turbulent region. This flow setup allows investigation of the initial transition and statistically steady stages of the interaction. Shock deformation is quantified using the local shock wave position. The root-mean-square (rms) fluctuation in the shock wave position increases during the initial stage of the interaction, for which the time interval divided by the integral time scale increases with Mt2/(Ms2−1), where Mt is a turbulent Mach number and Ms is a shock Mach number. In late time, the rms fluctuation in the shock wave position hardly depends on the propagation time and follows a power law, [Mt2/(Ms2−1)]0.46, whose exponent is similar to the power law exponent of the rms pressure-jump fluctuation reported in experimental studies. Fluctuations in the shock wave position have a Gaussian probability density function. The spectral analysis confirms that the length scale that characterizes shock wave deformation is the integral length scale of turbulence. The fluctuating shock wave position is correlated with dilatation of the shock wave, where the correlation coefficient increases with Mt/(Ms − 1). In addition, the shock wave that deforms backward tends to be stronger than average and vice versa. Mean pressure jumps across the shock wave are different between areas with forward and backward deformations. This difference increases with the rms fluctuation in the shock wave position and is well-represented as a function of Mt2/(Ms2−1).
Interaction of a shock wave with turbulence can generate large thermodynamic fluctuations in a flow. This can lead to enhanced mixing, peak heat transfer, and high sound level. The canonical interaction of homogeneous isotropic turbulence with a nominally normal shock wave acts as a model problem to investigate physics and develop predictive models. The case of purely vortical turbulence upstream of the shock is arguably the most fundamental case, and it is the focus of the current work. Direct numerical simulation (DNS) data and linear interaction analysis (LIA) are used to develop a predictive model for the thermodynamic field. Specifically, transport-equation-based models are proposed for the variances in temperature, pressure, density, and entropy. The jump in the thermodynamic variances is modeled in terms of the mean compression at the shock, and the closure coefficients are obtained via Kovásznay mode decomposition. By comparison, the downstream decay is modeled in a phenomenological way in terms of acoustic transient near the shock and a far-field decay due to viscous dissipation. The model predictions are found to match well with available DNS data for a range of shock strengths. In addition, the closed-form solution of the model equations gives the scaling of the thermodynamic fluctuations with mean flow Mach number.
Interactions between a spherical shock wave and a turbulent cylinder wake are studied with wind tunnel experiments. The shock wave is generated outside the wake and propagates across the turbulent wake. Instantaneous streamwise velocity is measured on the wake centerline while peak overpressure of the shock wave is measured outside the wake after the shock wave has passed across the wake. The experiments are performed for various conditions of the cylinder wake to investigate the influences of the root-mean-squared (rms) velocity fluctuation and of the length of the turbulent region through which the shock wave propagates. The velocity fluctuation opposite to the shock propagation direction is positively correlated with the peak-overpressure fluctuation. The mean peak overpressure decreases after the shock wave propagates in the wake. These relations between velocity and peak overpressure are explained by the shock-surface deformation, where the peak overpressure is increased and decreased, respectively, for the shock surfaces with concave and convex shapes in relation to the shock propagation direction. The correlation coefficients between the velocity and peak-overpressure fluctuations and the rms peak-overpressure fluctuation increase with the rms velocity fluctuation. The rms peak-overpressure fluctuation becomes independent of the turbulent length on the shock ray once the shock wave has propagated through a sufficiently long turbulent region. The peak-overpressure fluctuation has a probability density function (PDF) close to a Gaussian shape even though the PDF of velocity fluctuations in the wake is negatively skewed.
The overpressure fluctuations behind a weak shock wave interacting with turbulence are studied by wind tunnel experiments, where a spherical shock wave propagates in grid turbulence. The experiments are conducted for various values of the shock Mach number MS0 of the shock wave and turbulent Mach number MT of the grid turbulence. The experimental results show that the root-mean-squared peak-overpressure fluctuation divided by the averaged peak-overpressure, σΔp/⟨Δp⟩, where the inherent noise caused by the experimental facility is removed, follows a power law of MT2/(MS02−1). The probability density functions of the overpressure fluctuations are close to the Gaussian profile for a wide range of MT2/(MS02−1). A shock deformation model based on the deformation due to nonuniform fluid velocity is proposed for the investigation of the influences of turbulence on the shock wave. The deformation changes the cross-sectional area of the ray tube, which is related to the shock Mach number fluctuation of the area. The model for a weak shock wave yields the relation σΔp/⟨Δp⟩≈(1/3)[MT2/(MS02−1)]1/2, which agrees well with the experimental results. The model also predicts the Gaussianity of the peak-overpressure fluctuations behind the shock wave interacting with Gaussian velocity fluctuations. Good agreements between the model and experiments imply that the change in the shock wave characteristics by the interaction with turbulence is closely related to the shock wave deformation caused by the fluctuating turbulent velocity field.
The impacts of the interaction with grid turbulence of a turbulent Mach number in the range 0.6×10−2 to 2.4×10−2 (representative value) on a planar shock wave with a shock Mach number of about 1.04 were experimentally investigated. Using the counter-driver shock tube with a 120 mm × 120 mm square cross section, the turbulent Mach number level and the interaction length at the observation section were varied independently. Within resolvable experimental data, the shock Mach number became a decreasing function of the interaction length. The thickness of the shock front side-view profile increased as the turbulent Mach number and the interaction length increased. With a turbulent Mach number of 2.4×10−2 (representative value) and an interaction length longer than 200 mm, the shock front profile became fragmented. The condition for the fragmentation is consistent with the “broken shock” criterion proposed by numerical studies.
The non-dimensional dissipation rate C_{\unicode[STIX]{x1D700}}=\unicode[STIX]{x1D700}L/u^{\prime 3} , where \unicode[STIX]{x1D700} , L and are the viscous energy dissipation rate, integral length scale of turbulence and root-mean-square of the velocity fluctuations, respectively, is computed and analysed within the turbulent/non-turbulent interfacial (TNTI) layer using direct numerical simulations of a planar jet, mixing layer and shear free turbulence. The TNTI layer that separates the turbulent and non-turbulent regions exists at the edge of free shear turbulent flows and turbulent boundary layers, and comprises both the viscous superlayer and turbulent sublayer regions. The computation of C_{\unicode[STIX]{x1D700}} is made possible by the introduction of an original procedure, based on local volume averages within spheres of radius r , combined with conditional sampling as a function of the location with respect to the TNTI layer. The new procedure allows for a detailed investigation of the scale dependence of several turbulent quantities near the TNTI layer. An important achievement of this procedure consists in permitting the computation of the turbulent integral scale within the TNTI layer, which is shown to be approximately constant. Both the non-dimensional dissipation rate and turbulent Reynolds number Re_{\unicode[STIX]{x1D706}} vary in space within the TNTI layer, where two relations are observed: C_{\unicode[STIX]{x1D700}}\sim Re_{\unicode[STIX]{x1D706}}^{-1} and C_{\unicode[STIX]{x1D700}}\sim Re_{\unicode[STIX]{x1D706}}^{-2} . Specifically, whereas the viscous superlayer and part of the turbulent sublayer display C_{\unicode[STIX]{x1D700}}\sim Re_{\unicode[STIX]{x1D706}}^{-2} , the remaining of the turbulent sublayer exhibits C_{\unicode[STIX]{x1D700}}\sim Re_{\unicode[STIX]{x1D706}}^{-1} , which is consistent with non-equilibrium turbulence (Vassilicos, Annu. Rev. Fluid Mech. vol. 47, 2015, pp. 95–114).
Turbulent/non-turbulent interfaces (TNTIs) in compressible jets are studied with direct numerical simulations of temporally evolving compressible planar jets with jet Mach numbers MJ of 0.6, 1.6, and 2.6 ejected with a jet initial pressure equal to the ambient pressure. The flow properties near the TNTI are investigated with statistics computed on the local interfacial coordinate. The layer thicknesses are about 10-13η for the TNTI layer, 3η for the viscous superlayer, and 7-10η for the turbulent sublayer (TSL), where η is the Kolmogorov scale on the jet centerline. The TSL thickness divided by η decreases from 10 to 7 as MJ increases. The turbulent fluid is characterized with lower density, higher temperature, and lower pressure than the non-turbulent fluid, where these properties sharply change within the TNTI layer. The rate of change in internal energy near the TNTI is proportional to the initial kinetic energy of the jet, where the internal energy at the outer edge of the TNTI layer changes because of the diffusive/dilatational effects. The movement of entrained fluid is similar in compressible and incompressible jets. Compressibility affects the total entrainment rate via the total surface area of the TNTI, where the surface area of the TNTI per unit area of the plane perpendicular to the cross-streamwise direction decreases from 9.5 to 7.0 as MJ increases. Strongly compressive waves appear in the non-turbulent region at a high Mach number, where the imprints of these waves are found within the TNTI layer as strong pressure/temperature correlation and large values of pressure skewness.
We describe the critical condition necessary for the inner cylinder radius of a rotating detonation engine (RDE) used for in-space rocket propulsion to sustain adequate thruster performance. Using gaseous C2H4 and O2 as the propellant, we measured thrust and impulse of the RDE experimentally, varying in the inner cylinder radius r i from 31 mm (typical annular configuration) to 0 (no-inner-cylinder configuration), while keeping the outer cylinder radius (r o = 39 mm) and propellant injector position (r inj = 35 mm) constant. In the experiments, we also performed high-speed imaging of self-luminescence in the combustion chamber and engine plume. In the case of relatively large inner cylinder radii (r i = 23 and 31 mm), rotating detonation waves in the combustion chamber attached to the inner cylinder surface, whereas for relatively small inner cylinder radii (r i = 0, 9, and 15 mm), rotating detonation waves were observed to detach from the inner cylinder surface. In these small inner radii cases, strong chemical luminescence was observed in the plume, probably due to the existence of soot. On the other hand, for cases where r i = 15, 23, and 31 mm, the specific impulses were greater than 80% of the ideal value at correct expansion. Meanwhile, for cases r i = 0 and 9 mm, the specific impulses were below 80% of the ideal expansion value. This was considered to be due to the imperfect detonation combustion (deflagration combustion) observed in small inner cylinder radius cases. Our results suggest that in our experimental conditions, r i = 15 mm was close to the critical condition for sustaining rotating detonation in a suitable state for efficient thrust generation. This condition in the inner cylinder radius corresponds to a condition in the reduced unburned layer height of 4.5-6.5.
A method for predicting sonic boom waveforms emanating from a vehicle flying at supersonic speed is presented and compared to flight-test data. The prediction is achieved through three main steps: searching for the position of the sound source along the flight path, predicting the near-field waveform by a combination of computational fluid dynamics simulations on structured and unstructured meshes, and propagating this waveform to the far field by a numerical solver of the augmented Burgers equation. It is shown that this framework successfully predicts sonic boom signatures observed during the large-scale first phase of the drop test for simplified evaluation of the nonsymmetrically distributed sonic boom field test recently conducted by our research group at the Japan Aerospace Exploration Agency for both conventional N waves and lower-amplitude flattop waveforms.
We study the pressure increase across a planar shock wave with shock Mach numbers Ms of 1.1, 1.3, and 1.5 propagating through homogeneous isotropic turbulence at a low turbulent Mach number (Mt ∼ 10⁻⁴) based on direct numerical simulations (DNSs). Fluctuation in the pressure increase, Δp′, on a given shock ray is induced by turbulence around the ray. A local amplification of the shock wave strength, measured with the pressure increase, is caused by the velocity fluctuation opposed to the shock wave propagating direction with a time delay, while the velocity in the opposite direction attenuates the shock wave strength. The turbulence effects on the shock wave are explained based on shock wave deformation due to turbulent shearing motions. The spatial distribution of Δp′ on the shock wave has a characteristic length of the order of the integral scale of turbulence. The influence of turbulent velocity fluctuation at a given location on Δp′ becomes most significant after the shock wave propagates from the location for a distance close to the integral length scale for all shock Mach numbers, demonstrating that the shock wave properties possess strong memory even during the propagation in turbulence. A lower shock Mach number Ms results in a smaller rms value of Δp′, stronger influences on Δp′ by turbulence far away from the shock ray, and a larger length scale in the spatial profile of Δp′ on the shock wave. Relative intensity of Δp′ increases with [Mt/(Ms−1)]α, where DNS and experimental results yield α ≈ 0.73.
The scaling of the turbulent/non-turbulent interface (TNTI) at high Reynolds numbers is investigated by using direct numerical simulations (DNS) of temporal turbulent planar jets (PJET) and shear free turbulence (SFT), with Reynolds numbers in the range 142\leqslant Re_{\unicode[STIX]{x1D706}}\leqslant 400 . For Re_{\unicode[STIX]{x1D706}}\gtrsim 200 the thickness of the TNTI ( \unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D714}} ), like that of its two sublayers – the viscous superlayer (VSL, \unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D708}} ) and the turbulent sublayer (TSL, \unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D70E}} ) – all scale with the Kolmogorov micro-scale \unicode[STIX]{x1D702} , while the particular scaling constant depends on the sublayer. Specifically, for Re_{\unicode[STIX]{x1D706}}\gtrsim 200 while the VSL is always of the order of \unicode[STIX]{x1D702} , with 4\leqslant \langle \unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D708}}\rangle /\unicode[STIX]{x1D702}\leqslant 5 , the TSL and the TNTI are typically equal to 10\unicode[STIX]{x1D702} , with 10.4\leqslant \langle \unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D70E}}\rangle /\unicode[STIX]{x1D702}\leqslant 12.5 , and 15.4\leqslant \langle \unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D714}}\rangle /\unicode[STIX]{x1D702}\leqslant 16.8 , respectively.
In this study, we performed a direct numerical simulation (DNS) of a spatially developing shear mixing layer covering both developing and developed regions. The aim of this study is to clarify the driving mechanism and the vortical structure of the partial counter-gradient momentum transport (CGMT) appearing in the quasi self-similar region. In the present DNS, the self-similarity is confirmed in x/L ≥ 0.67 (x/δU0 ≥ 137), where L and δU0 are the vertical length of the computational domain and the initial momentum thickness, respectively. However, the trend of CGMT is observed at around kδU = 0.075 and 0.15, where k is the wavenumber, δU is the normalized momentum thickness at x/L = 0.78 (x/δU0 = 160), and kδU = 0.075 corresponds to the distance between the vortical/stretching regions of the coherent structure. The budget analysis for the Reynolds shear stress reveals that it is caused by the pressure diffusion term at the off-central region and by −p(∂u/∂y)¯ in the pressure-strain correlation term at the central region. As the flow moves toward the downstream direction, the appearance of those terms becomes random and the unique trend of CGMT at the specific wavenumber bands disappears. Furthermore, we investigated the relationship between the CGMT and vorticity distribution in the vortex region of the mixing layer, in association with the spatial development. In the upstream location, the high-vorticity region appears in the boundary between the areas of gradient momentum transport and CGMT, although the high-vorticity region is not actively producing turbulence. The negative production area gradually spreads by flowing toward the downstream direction, and subsequently, the fluid mass with high-vorticity is transported from the forehead stretching region toward the counter-gradient direction. In this location, the velocity fluctuation in the high-vorticity region is large and turbulence is actively produced. In view of this, the trend of negative production appears in the flow where the turbulence production and non-turbulent regions mix. Then, the non-turbulent region and CGMT almost simultaneously disappear in the fully developed region.
In this study, simultaneous measurement of velocity and temperature in a heated planar jet, which has a Reynolds number of 22,000 based on the nozzle height and no co-flow, is performed to investigate the evolution of the scalar field related to the coherent structures in the jet. A combined probe, which consists of a hot-wire and a cold-wire, is developed and used for simultaneous measurement. The cold-wire is placed 2.0 mm upstream of the hot-wire to keep the measurement accuracy of the hot-wire caused by the streamwise velocity fluctuation by the cold-wire. The results of POD analysis show that the shapes of the eigenfunctions of heat flux are different at each downstream location, and they are also different from those of velocity and temperature. They have peak values at a position in-between two peak values of the eigenfunctions of the first mode for velocity and temperature. Further, the position is consistent with that of the peak values in the cross-streamwise profile of turbulent heat flux. From these results, it can be concluded that the turbulent heat flux field can be expressed not only by the dominant modes of the velocity or temperature fields but also as a combination of both.
We perform direct numerical simulations of shock-wave/boundary-layer interactions (SBLI) at Mach number M = 1.7 to investigate the influence of the state of the incoming boundary layer on the interaction properties. We reproduce and extend the flow conditions of the experiments performed by Giepman et al., in which a spatially evolving laminar boundary layer over a flat plate is initially tripped by an array of distributed roughness elements and impinged further downstream by an oblique shock wave. Four SBLI cases are considered, based on two different shock impingement locations along the streamwise direction, corresponding to transitional and turbulent interactions, and two different shock strengths, corresponding to flow deflection angles 3 degreees and 6 degrees. We find that, for all flow cases, shock induced separation is not observed, the boundary layer remains attached for the 3 degrees case and close to incipient separation for the 6 degrees case, independent of the state of the incoming boundary layer. The findings of this work suggest that a transitional interaction might be the optimal solution for practical SBLI applications, as it removes the large separation bubble typical of laminar interactions and reduces the extent of the high-friction region associated with an incoming turbulent boundary layer.
The characteristics of divergence-free grid turbulence interacting with a weak spherical shock wave with a Mach number of 1.05 are experimentally investigated. Turbulence-generating grids are used to generate nearly isotropic, divergence-free turbulence. The turbulent Reynolds number based on the Taylor microscale Reλ and the turbulent Mach number Mt are 49≤Reλ≤159 and 0.709 × 10−3≤Mt≤2.803×10−3, respectively. A spherical shock wave is generated by a diaphragmless shock tube. The instantaneous streamwise velocity before and after the interaction is measured by a hot wire probe. The results show that the root-mean-square value of streamwise velocity fluctuations (r.m.s velocity) increases and the streamwise integral length scale decreases after the interaction. The changes in the r.m.s velocity become small with the increase in Reλ and Mt for the same strength of the shock wave. This tendency is similar to that of the streamwise integral length scale. The continuous wavelet analysis shows that high intensity appears mainly in the low-frequency region and positive and negative wavelet coefficients appear periodically in time before the interaction, whereas such high intensity appears in both the low- and high-frequency regions after the interaction. The spectral analysis reveals that the energy at high wavenumbers increases after the interaction. The change in turbulence after the interaction is explained from the viewpoint of the initial turbulent Mach number. It is suggested that the change is more significant for initial divergence-free turbulence than for curl-free turbulence.
Response time of the post-shock wave (SW) overpressure modulation by turbulence is investigated in wind tunnelexperiments. A peak-overpressure fluctuation, observed on a wall, is induced by turbulence around the SW ray, but away from the wall, demonstrating finite response time of the modulation. We propose a model of the modulation based on the SW deformation by a local flow disturbance, which yields the response time being proportional to the product of the large-eddy turnover time and (MT/MS0)0.5 (MT: turbulentMach number and MS0: shock Mach number), in consistent with the experiments.
The interactions between homogeneous turbulence and a planar shock wave are analytically investigated using rapid distortion theory (RDT). Analytical solutions in the solenoidal modes are obtained. Qualitative answers to unsolved questions in a report by Andreopoulos et al. ( Annu. Rev. Fluid Mech. , vol. 524, 2000, pp. 309–345) are provided within the linear theoretical framework. The results show that the turbulence kinetic energy (TKE) is increased after interaction with a shock wave and that the contributions to the amplification can be interpreted primarily as the combined effect of shock-induced compression, which is a direct consequence of the Rankine–Hugoniot relation, and the nonlinear effect, which is an indirect consequence of the Rankine–Hugoniot relation via the perturbation manner. For initial homogeneous axisymmetric turbulence, the amplification of the TKE depends on the initial degree of anisotropy. Furthermore, the increase in energy at high wavenumbers is confirmed by the one-dimensional spectra. The enstrophy is also increased; its increase is more significant than that of the TKE because of the significant increase in enstrophy at high wavenumbers. The vorticity components perpendicular to the shock-induced compressed direction are amplified more than the parallel vorticity component. These results strongly suggest that a high resolution is needed to obtain accurate results for the turbulence–shock-wave interaction. The integral length scales ( L ) and the Taylor microscales ( ) are decreased for most cases after the interaction. However, and are amplified. Here, the subscripts 2 and 3 indicate the perpendicular components relative to the shock-induced compressed direction. The dissipation length and TKE dissipation rate are amplified.
Direct numerical simulations (DNS) of temporally evolving shear layers have been performed to study the entrainment of irrotational flow into the turbulent region across the turbulent/non-turbulent interface (TNTI). Four cases with convective Mach number from 0.2 to 1.8 are used. Entrainment is studied via two mechanisms; nibbling, considered as vorticity diffusion across the TNTI, and engulfment, the drawing of the pockets of the outside irrotational fluid into the turbulent region. The mass flow rate due to nibbling is calculated as the product of the entrained mass flux with the surface area of the TNTI. It is found that increasing the convective Mach number results in a decrease of the average entrained mass flux and the surface area of the TNTI. For the incompressible shear layer the local entrained mass flux is shown to be highly correlated with the viscous terms. However, as the convective Mach number increases, the mass fluxes due to the baroclinic and the dilatation terms play a more important role in the local entrainment process. It is observed that the entrained mass flux is dependent on the local dilatation and geometrical shape of the TNTI. For a compressible shear layer, most of the entrainment of the irrotational flow into the turbulent region due to nibbling is associated with the compressed regions on the TNTI. As the convective Mach number increases, the percentage of the compressed regions on the TNTI decreases, resulting in a reduction of the average entrained mass flux. It is also shown that the local shape of the interface, looking from the turbulent region, is dominated by concave shaped surfaces with radii of curvature of the order of the Taylor length scale. The average entrained mass flux is found to be larger on highly curved concave shaped surfaces regardless of the level of dilatation. The mass fluxes due to vortex stretching, baroclinic torque and the shear stress/density gradient terms are weak functions of the local curvatures on the TNTI, whereas the mass fluxes due to dilatation and viscous diffusion plus the viscous dissipation terms have a stronger dependency on the local curvatures. As the convective Mach number increases, the probability of finding highly curved concave shaped surfaces on the TNTI decreases, whereas the probability of finding flatter concave and convex shaped surfaces increases. This results in a decrease of the average entrained mass flux across the TNTI. Similar to the previous works on jets, the results show that the contribution of the engulfment to the total entrainment is small for both incompressible and compressible mixing layers. It is also shown that increasing the convective Mach number decreases the velocities associated with the entrainment, i.e. induced velocity, boundary entrainment velocity and local entrainment velocity.
The noise problem associated with an aircraft flying at supersonic speeds is shown to depend primarily on the shock wave pattern formed by the aircraft. The noise intensity received by a ground observer from a supersonic aircraft flying at high as well as low altitudes, is shown to be high, although it is of a transient nature.
A study of the shock wave patterns around an aircraft in accelerated and retarded flight is shown to lead to an explanation of the one or more booms, of short duration, heard by ground observers after an aircraft has dived at supersonic speeds.
The shock wave patterns associated with an aircraft flying in accelerated or retarded flight at transonic speeds are shown in certain cases to be very different from the corresponding patterns observed in steady flight. The significance of these results, with reference to problems of flight at supersonic speeds is briefly discussed.
When a shock wave ejected from the exit of a 5.4-mm inner diameter, stainless steel tube propagated through grid turbulence across a distance of 215 mm, which is 5–15 times larger than its integral length scale
, and was normally incident onto a flat surface; the peak value of post-shock overpressure,
, at a shock Mach number of 1.0009 on the flat surface experienced a standard deviation of up to about 9 % of its ensemble average. This value was more than 40 times larger than the dynamic pressure fluctuation corresponding to the maximum value of the root-mean-square velocity fluctuation,
. By varying
and
, the statistical behavior of
was obtained after at least 500 runs were performed for each condition. The standard deviation of
due to the turbulence was almost proportional to
. Although the overpressure modulations at two points 200 mm apart were independent of each other, we observed a weak positive correlation between the peak overpressure difference and the relative arrival time difference.
The interaction between isotropic turbulence and a normal shock wave is investigated through a series of direct numerical simulations at different Reynolds numbers and mean and turbulent Mach numbers. The computed data are compared to experiments and linear theory, showing that the amplification of turbulence kinetic energy across a shock wave is described well using linearized dynamics. The post-shock anisotropy of the turbulence, however, is qualitatively different from that predicted by linear analysis. The jumps in mean density and pressure are lower than the non-turbulent Rankine–Hugoniot results by a factor of the square of the turbulence intensity. It is shown that the dissipative scales of turbulence return to isotropy within about 10 convected Kolmogorov time scales, a distance that becomes very small at high Reynolds numbers. Special attention is paid to the ‘broken shock’ regime of intense turbulence, where the shock can be locally replaced by smooth compressions. Grid convergence of the probability density function of the shock jumps proves that this effect is physical, and not an artefact of the numerical scheme.
We report an experimental analysis of the local entrainment velocity in the self-similar region of a turbulent jet. Particle tracking velocimetry is performed to determine the position of the convoluted, instantaneous turbulent/non-turbulent interface and to compute velocity and velocity derivatives in the proximity of the interface. We find that the local entrainment velocity is mostly governed by a viscous component and that its magnitude depends on the local shape of the interface. It is illustrated that local entrainment is faster for surface elements concave towards the turbulent region. A closer analysis of the plane spanned by mean and Gaussian curvature reveals that depending on the surface shape, different small-scale mechanisms are dominant for the local entrainment process, namely, viscous diffusion for concave shapes and vortex stretching for convex shapes. Key quantities influencing viscous diffusion and vortex stretching in the entrainment process are identified. It is illustrated that the viscous advancement of the interface into the non-turbulent region mostly depends on the shape of the enstrophy profile normal to the interface. The inviscid contribution is intimately related to the alignment of vorticity with the eigenvectors of the rate of strain tensor. Finally, the analysis substantiates that the convolution of the instantaneous interface is driven by the advection of the underlying fluid together with a contribution from the local entrainment velocity, with the advection velocity being the governing part.
Amplification of turbulent kinetic energy in an axial compression is examined in the frame of homogeneous rapid distortion theory (RDT) by using the Craya–Herring formalism. By separating the turbulent field into solenoidal and dilatational modes (Helmholtz decomposition), one can show the dilatational mode is mediated by the parameter Δm0=D0/a0k0, which corresponds to the initial ratio between the acoustic time scale (a0k0)−1 and the compression time scale D0−1, with D0 the compression rate. It is shown here that amplification of total kinetic energy is then limited by two analytical solutions obtained for Δm0=0 (purely solenoidal-acoustical regime) and for Δm0≫1 (‘‘pressure released’’ regime), respectively. The results of the theory are first compared to results of direct numerical simulations (DNS) on homogeneous axial compression. The applicability of this homogeneous approach to the shock wave turbulence interaction, is then discussed. Considering a shock-induced compression at given Mach number, it is shown that the corresponding amplification factors predicted by homogeneous RDT largely differs from that obtained from Ribner’s linear interaction analysis and DNS on the shock problem.
A jet flow was used to model roughly a localized region of atmospheric turbulence, simulating a single idealized “eddy.” The jet was arranged in the UTIAS 80‐ft sonic‐boom generator horn so as to blow either against or with the direction of boom propagation. The two cases produced spiked and rounded boom signatures, respectively, qualitatively in accord with theory. The resemblance to signatures resulting from supersonic flight under turbulent atmospheric conditions was especially marked with the spiked “superbooms.”
This paper reports on the extension of second-order models for unforced isotropic turbulence to the compressible regime. The equations contain two principal compressibility terms: the dilatation dissipation and the pressure–dilatation correlation. The dilatation dissipation has been modeled in accordance with the shocklet dissipation theory of Zeman [Phys. Fluids A 2, 178 (1990)]; the pressure–dilatation term has been identified with the rate of change of compressible potential energy represented by the pressure fluctuation variance ∼(p2). The rate of change of ∼(p2) is assumed to be governed by a nonlinear relaxation mechanism that drives ∼(p2) to an equipartition value. By comparison with direct numerical simulations of 2-D and 3-D turbulence, it is shown that the extended turbulence energy equations are capable of simulating the compressibility effects associated with rms Mach number and the (initial) pressure fluctuations.