The stratified Kelvin–Helmholtz instability (KHI) test case is a famous problem which manifests itself when there is a velocity difference at the interface between two fluids of different densities . It can commonly be observed through experimental observation and numerical simulation, and it is also visible in many natural phenomena, for example in situations with wind flow over bodies of water causing wave formation and in the planet Jupiter's atmosphere between atmospheric bands moving at different speeds . The study of this instability in a benchmark formulation reveals key information about the transition to turbulence for two fluids moving at different speeds. For these practical applications, it is common to choose a double shear layer problem to simulate the formation of KHI in a periodic two-dimensional computational setting with unit side length. This stratified shear layer instability problem is used to demonstrate the evolution of linear perturbations into a transition to nonlinear two-dimensional hydrodynamic turbulence. The instability triggers small-scale vortical structures at the sharp density interface initially, which eventually transitions through nonlinear interactions to a completely turbulent field.

A two-dimensional implementation of the dual-shear layer KHI problem is devised through our aforementioned unstable perturbed compressible shear layer. This may be implemented through our computational domain which is a square of unit side length with the following initial conditions: 5ρ(x,y)=1.0,  if  |y|≥0.252.0,  if  |y|<0.256u(x,y)=α,if  |y|≥0.25-α,if  |y|<0.257v(x,y)=λsin⁡(2πnx/L),8p(x,y)=2.5. We can observe that the vertical component of the velocity is perturbed using a single-mode sine wave (n=2, L=1) with an amplitude λ=0.01. Our two-dimensional numerical experiments are solved to a final dimensionless time of t=5. We clarify that the R2 simulation domain for all experiments is set in (x,y)∈[-0.5,0.5]×[-0.5,0.5] with N2=163842 degrees of freedom. Figure  represents a schematic expressing the initial conditions of our two-dimensional simulation. We remark that in this study we perform implicit large eddy simulation (ILES) simulations by using a finite-volume framework. Our numerical scheme utilizes the fifth-order accurate, weighted essential non-oscillatory (WENO) reconstructions equipped with Roe's approximate Riemann solver at the cell interfaces. It is well known that the utilization of the artificial dissipation mechanism of ILES schemes (from the numerical viscosity of upwind biased state reconstructions) mimics the physical viscosity of the Navier–Stokes equations in the limit of infinite Reynolds numbers. We utilize a parallel approach for the computational solution of our governing laws implemented in the OpenMPI framework. Details about the implementation and the computational performance of our solver may be found in , additionally showing weak and strong scaling tests. Our three-dimensional simulations employ a similar approach.

Figure  describes snapshots in time of the density field for this two-dimensional compressible turbulence test case when α=1.0. One can notice a transition to turbulence once an initial instability has developed. The shearing velocity magnitude given by α controls the compressibility which is apparent from comparisons with Figs.  and where smaller values lead to formation of much smoother structures and consequently lead to shock-free fields in the incompressible limit. Evidence from Fig.  also shows a delay in the onset of turbulence due to a reduced shearing velocity. Table  also demonstrates the mean and maximum Mach number values at the final computational time t=5. It is clear that the case for α=0.25 corresponds to a perfectly subsonic regime with lower compressibility (i.e., the mean Mach number of M=0.15).

Figure  demonstrates the time evolution characteristics of the 2-D KHI problem. On the left, we illustrate the time series of the domain-integrated velocity amplitude (i.e., the root mean square values of the kinetic velocity) normalized with its initial condition with each α value. It is clear that the KHI instability starts earlier for larger α values. We also demonstrate the evolution of the compensated kinetic energy spectrum on the right for α=1.0. Similar statistical trends are observed at each time. Therefore, we will only focus on the results at the final time t=5 in our statistical analysis presented in the next section.

While two-dimensional compressible turbulence investigations are valuable for insight into the physical processes of systems which exhibit extreme aspect ratios , it is well known that the process of energy transfer between scales is fundamentally different when compared to that of three-dimensional flows . Isotropic, homogeneous, incompressible three-dimensional turbulence is characterized by the famous Kolmogorov–Richardson cascade of energy where the largest vortices continuously inject energy into an inertial cascade which terminates in the Kolmogorov length scale where viscous effects dissipate this energy. This is particularly applicable for engineering flows, where it has been established that turbulence “decays” in the absence of forcing due to viscous dissipation. In contrast, two-dimensional turbulence exhibits the presence of an inverse energy cascade (given by Kraichnan–Batchelor–Leith theories; ) where energy from the smallest scales is transferred to the largest scales. This has implications for the restoration of local isotropy (since large-scale structures created by the inverse energy cascade affect the amount of enstrophy in the field and thus affect the energy dissipation rate). In the presence of periodic boundary conditions (a subject of future investigations), these newly created large-scale structures may lead to significant alteration in scaling laws.

Our computational domain for the three-dimensional turbulence case is analogous to that of the two-dimensional domain. We utilize a domain given by a R3 set in (x,y,z)∈[-0.5,0.5]×[-0.5,0.5]×[-0.5,0.5] with N3=5123 degrees of freedom. Our initial conditions are given by 9ρ(x,y,z)=1.0,  if  |y|≥0.252.0,  if  |y|<0.2510u(x,y,z)=α,if  |y|≥0.25-α,if  |y|<0.2511v(x,y,z)=λsin⁡(2πnx/L),12w(x,y,z)=λsin⁡(2πnz/L),13p(x,y,z)=2.5, and periodic boundary conditions in all directions. We keep our parameters n, L and λ similar to those used in the two-dimensional case and utilize N3=5123 degrees of freedom for the simulation of our computational domain.

Figure  shows the density field at times t=1 and t=5 for a shearing velocity magnitude of α=1.0. One can observe how the solution domain has transitioned almost entirely to a turbulent field for this case as against the very visible stratification observed in lower compressibility simulations given by α=0.5 and α=0.25 shown in Figs.  and , respectively. Our aim is to quantify the effect of the shearing velocity on the compressibility and scaling laws of these co-designed two- and three-dimensional configurations. Similar to the two-dimensional case, we have plotted the time evolution of the domain-integrated velocity in Fig.  between t=0 and t=5. The decay rates in three-dimensional simulations are substantially higher than those obtained in two-dimensional simulations. This can be attributed to the use of a lesser number of grid points sampled in each direction. However, the energy spectrum trend is similar and yields a k-5/3 spectrum at each time. In the following section, we thus present a systematic analysis based on data obtained at t=5.

The first statistical measure we investigate is given by the classical kinetic energy spectra. To obtain these spectra, we start with an expression for the spatial kinetic energy in wavenumber space given by 14E(k,t)=12|u^(k,t)|2, where u^(k,t) is the Fourier transform of the velocity vector in the wavenumber space. Equation () can also be rewritten in terms of velocity components (assuming a two-dimensional Cartesian domain) as 15E(k,t)=12|u^(k,t)|2+|v^(k,t)|2, where we compute velocity components u^(k,t) and v^(k,t) using a fast Fourier transform algorithm . Finally, the spectra can be calculated by integrating over a unit bandwidth (i.e., angle-averaged) in the following manner: 16E(k,t)=∑k-12≤|k′|<k+12E(k′,t), where k=|k|=kx2+ky2 in R2. Extensions to three dimensions are straightforward.

The kinetic energy spectrum is generally utilized for characterizing the energy content of scales in incompressible turbulent flows and does not take the localized scale content of the density into consideration. To include these density effects, following and , we define an energy spectrum built on density-weighted velocity ω=ρu, i.e., through using 17E(k,t)=12|ω^(k,t)|2, where we can apply the same angle-averaged rule given by Eq. () to obtain one-dimensional spectra.

Figure  describes the spherical-averaged energy spectra for the three-dimensional test case. Note here that the spherical average implies that the local energy content in the Fourier domain is integrated over a spherical shell of radius k in three dimensions. One can observe a scaling behavior that corresponds to classical Kolmogorov theory in the infinite Reynolds number limit (i.e., an inertial range with k-5/3 scaling) for both purely kinetic energy spectra and density-weighted kinetic energy spectra. The finer dissipative scales are seen to display a k-6 scaling behavior for both these statistical quantities as well. We have also plotted the compensated energy spectra, which illustrate the scaling laws more quantitatively following the horizontal lines.

The data presented in Fig.  have been obtained by performing a three-dimensional fast Fourier transform (FFT) procedure. From a practical implementation point of view, we perform a slightly different approach to compute energy spectra. The main advantage of this procedure is that it is naturally suited to any parallel computing architecture. For an analogy with the two-dimensional test cases, we present transversely averaged energy spectra in Fig.  wherein the circular averaging of the energy in the Fourier domain is carried out over different two-dimensional z planes which are then spatially averaged over the depth of the domain. Similar trends to the spherical averaging spectral scaling are observed for this case. However, we note that the obtained spectra are less noisy when using a direct three-dimensional FFT procedure. This can be interpreted by the quasi-homogeneity of the flow after the onset of turbulence.

We investigate the performance of the same metrics for the two-dimensional test case and obtain scaling behavior as seen in Fig.  where a k-3 scaling behavior is obtained in accordance with the direct cascade of enstrophy espoused by Kraichnan–Batchelor–Leith (KBL) theory for the inertial range, especially for the lower compressibility ratio. A higher magnitude of α is seen to yield a more flattened spectrum towards k-7/3 scaling and also delay the formation of the k-6 cascade in the dissipation range. Figure  also shows the spectral scaling obtained from the density-weighted kinetic energy spectra where scaling behavior corresponding to k-7/3 is seen for all α values. This suggests that the two-dimensional configuration of the test case is affected by the packaging of density content at different scales. The dissipation zone shows a similar behavior using this metric where a delay in scaling with k-6 is obtained by an increase in the magnitude of α. We can conclude that the density-weighted spectrum becomes a more universal representation for various degrees of compressibility.

Figure  shows the effect of the parameter α on the compressibility of the two-dimensional turbulence case through the use of compensated energy spectra where the distance from the origin in the Fourier space (in other words k) is used to weight instantaneous energy content. We only present the compensated energy distribution in the first quadrant of the Fourier space. At α=1.0 one can observe a distinct loss of isotropy in the energy content of the solution field (in spectral space) which corresponds to an enhanced compressibility. In comparison, α=0.5 and α=0.25 display a behavior which is rather isotropic in nature, indicating weak compressibility.

To demonstrate the effect of density more clearly, we present the difference spectra for the 2-D KHI turbulence in Fig. . Here, we compute the spectrum of the difference between the velocity u and the normalized density-weighted velocity ρu/〈ρ〉, where ρ refers to the spatial average of the square root of density. The results show a clear inertial range with the k-5/3 scaling. This is a manifestation of the density effect in 2-D KHI turbulence.

To study the effect of compressibility in more detail we perform the Helmholtz decomposition to compute energy spectra from the curl-free and divergence-free components of the velocity field. This decomposition has been extensively used in turbulence studies (i.e., see ). In our present work, we investigate the behavior of energy spectra using both the kinematic velocity and density-weighted velocity fields in 2-D and 3-D KHI turbulence problems. Let v be a vector field in ∈Rn (e.g., v could be the kinetic velocity field u or the density-weighted velocity field ω=ρu); then, v can be decomposed into a curl-free component and a divergence-free component : 18v=∇ϕ+∇×A, which can be rewritten as 19v=vc+vs, where vc=∇ϕ is the compressive (curl-free) component since the curl of a gradient of any scalar field ϕ is zero, and vs=∇×A is the solenoidal (divergence-free) component since the divergence of a curl of any vector field A is zero. Taking the divergence of Eq. () yields the following Poisson equation: 20∇⋅v=∇2ϕ, which can be solved for ϕ efficiently using an FFT procedure since v is provided as a quantity of interest that we would like to decompose into two parts. Once ϕ is computed, the compressive and solenoidal parts can be easily computed as follows: 21vc=∇ϕ,22vs=v-vc. We note that there would be infinitely many candidates for the compressive component since the multiplication of ϕ by any arbitrary constant after solving the Poisson equation would still yield a curl-free velocity field. However, the energy spectrum scaling behaviors would remain identical for each realization.

Figure  presents the compensated energy spectra for the 3-D KHI problem using both definitions of the velocity vector field (i.e., the kinematic velocity and the density-weighted velocity). We have obtained a k-5/3 dominant scaling for the solenoidal component in both definitions. However, the compressive component demonstrates an anomalous spectrum especially when we use the kinetic velocity definition. This anomaly can also be linked to the results of the pressure power spectra that we present in the next section. Figure  presents the same analysis for the case of 2-D KHI turbulence. Both compressive and solenoidal components scale with the k-5/3 slope for the density-weighted velocity field. However, there is a clear difference for the results with various values of α when we look at the Helmholtz decomposition of the kinetic velocity field. The solenoidal inertial range scaling becomes k-3 for lower α values, which is consistent with Kraichnan theory. However, the scaling steepens and gets closer to k-2 for increasing α, which is also consistent with the Kadomtsev–Petviashvili spectrum for acoustic turbulence.

Observations on the density power spectrum have played an important role in astrophysics applications . Although it has been established that the density power spectrum has an inertial scaling of k-5/3 , similar to the Kolmogorov energy spectrum, demonstrated that it depends on the flow regime as well as the initial conditions by considering a three-dimensional weakly compressible hydrodynamic turbulence setup. By studying weakly compressible two-dimensional flows, showed that the density spectrum scales between k-1 and k-5 for nonuniform and uniform entropy cases, respectively. They presented a great discussion for state-of-the-art computations and scaling law observations for the density power spectrum.

In order to quantify the effect of the scale content of density alone, we devise a power spectrum that reflects the average packaging of density over different scales at any given time in the simulation. This may be given by the following expression: 23Γ(k,t)=12|ρ^(k,t)|2, followed by angle averaging which leads to 24Γ(k,t)=∑k-12≤|k′|<k+12Γ(k′,t).

Observations regarding the difference in scaling behavior of the kinetic energy and density-weighted kinetic energy spectra give us a cause to compare the scaling behavior of the density power spectra for both our two- and three-dimensional test cases. Figure  shows the density power spectra for the three-dimensional turbulence test case where it can be seen that a five-thirds law is followed for the arrangement of density content in the solution field. A dissipation range scaling of k-6 can also be observed. It can be seen that the variation of parameter α does not seem to affect scaling behavior appreciably. Figure  shows a similar examination for the two-dimensional test case where a considerable difference in scaling behavior is observed. The imposition of two-dimensional turbulence leads to a considerable alteration in the scaling behavior of the density power spectrum with a k-5/3 scaling observed in the inertial range and a k-3 scaling in the dissipation range. In fact, this packaging of density consequently affects the density-weighted kinetic energy spectra described in Fig. . The intercomparison of the two- and three-dimensional statistical quantities suggests that the density power spectrum (i.e., the arrangement of density at different wavenumbers) plays an important role with increased compressibility of any simulation wherein the k-5/3 scaling causes a deviation from k-3 scaling associated with two-dimensional incompressibility to k-7/3 scaling for α=1.0 for the same test case. In contrast, the k-5/3 density power spectrum of three-dimensional turbulence causes no variation in scaling behavior with increased compressibility and also causes similar scaling behaviors for both averaged kinetic energy spectra as well as averaged density-weighted kinetic energy spectra as seen in Fig. . This is one of the central conclusions of this investigation.

Similar to the density power spectrum defined in Eq. (), the pressure power spectrum can be computed as 25Π(k,t)=12|p^(k,t)|2, and its angle-averaged form reads as 26Π(k,t)=∑k-12≤|k′|<k+12Π(k′,t).

As discussed in , the pressure spectrum can be expressed by Π(k)∝kE(k)2 by considering dimensional arguments. Indeed, this yields a pressure spectra scaling of k-7/3 for the Kolmogorov regime and a pressure spectra scaling of k-5 for the Kraichnan regime. Figures  and demonstrate the pressure power spectra for the 3-D and 2-D KHI problems, respectively. In the 3-D case, it is clear that our results are consistent with the theoretical estimate of k-7/3 scaling for all values of the compressibility parameter α. However, in 2-D turbulence we only observe k-5 scaling for smaller scales (i.e., higher wavenumbers). Particularly for weaker compressibility, given by the α=0.25 case, the k-5 scaling starts earlier. Figure  clearly illustrates that the pressure power spectrum inertial scaling becomes k-5/3 for stronger compressibility. These results indicate that the pressure power spectrum can be a useful tool for characterizing two-dimensional compressible turbulence.

Statistical inferences about the nature of compressible turbulence may also be drawn through the use of velocity structure functions which also show scaling tendencies according to the physics of the solution field . A velocity structure function may be expressed as 27Sp(r)=〈(u(x+r)-u(x))p〉, where the ensemble averaging is taken over all positions x and all orientations of r within the computational domain to yield statistics for the length scale r=|r|. Our choice of p determines the order of the structure function we are examining and this investigation looks at p=2 for the characterization of turbulence in both two and three dimensions. The second-order structure function has been used to characterize the turbulence in both 2-D (e.g., see ) and 3-D (e.g., see ) turbulent flows. We note that some researchers have preferred to use the absolute value definition, which might change the results for odd values of p (e.g., see , for a great discussion on various definitions of the structure functions). For the 2-D turbulence setting, predicted a scaling law of rn-1 where n refers to the scaling component of the energy spectrum (i.e., E(k)∝k-n). In 3-D turbulence, the scaling of rp/3 has been established for the pth structure function. Both longitudinal (u∥r) and transverse (u⟂r) third-order velocity structure functions are computed in the present study. In our assessments, a range of 10-2≤r≤10-1 is assumed to represent the general vicinity of the inertial range.

We utilize the high-fidelity data of the previously described numerical experiments for two- and three-dimensional turbulence to obtain structure function statistics at time t=5. Figure  shows the second-order velocity structure function for the longitudinal and transverse directions for the 3-D test case. One can observe a steadily increasing alignment with r2/3 with a decreasing value of α, implying weaker compressibility. It is worth mentioning here that Kolmogorov theory dictates a cascade given by p/3. Similar trends are observed for both longitudinal and transverse directions, suggesting that a certain degree of isotropy now characterizes the system. For ranges of r below 10-2, it is observed that both longitudinal and transverse structure functions scale according to r2 for the second-order structure function.

We undertake a similar statistical examination for our two-dimensional test case where second-order longitudinal and transverse structure functions are given by Fig. , where it is observed that at low r, a scaling corresponding to r2 is observed. This is in accordance with findings in . At larger values of r, the r2 scaling transitions to a r4/3 scaling at relatively higher compressibility (i.e., α=1.0) and r scaling at α=0.25. Eventually, it is expected that an r2/3 behavior must emerge with perfect incompressibility. The aforementioned observations hold true for both longitudinal and transverse second-order structure functions and are consistent with the definition of S(r)∝rn-1. It can be observed that the velocity structure functions for three-dimensional simulations generally obey the prediction of the Kolmogorov theory (for lower values of α indicating weak compressibility) as against their two-dimensional counterparts.