Volume 21, Issue 2, February 2009

The impact of a solid sphere on a liquid surface has challenged researchers for centuries and remains of interest today. Recently, Duez et al. [Nat. Phys.3, 180 (2007)] published experimental results of the splash generated when a solid sphere enters water. Interestingly, the microscopic properties of the solid surface control the nature of the macroscopic behavior of the splash. So by a change in the surface chemistry of the solid sphere, a big splash can be turned into an inconspicuous disappearance and vice versa. This problem was investigated by numerical simulations based on the Navier–Stokes equations coupled with the Cahn–Hilliard equations. This system allows us to simulate the motion of an airwater interface as a solid sphere impacts the liquid pond. The inclusion of the surface energies of the solid surface in the formulation gives a reasonably quantitative description of the dynamic wetting. Numerical results with different wetting properties and impact speed are presented and directly compared with the recent experimental results from Duez et al.
 AWARD AND INVITED PAPERS


Twenty years of experimental and direct numerical simulation access to the velocity gradient tensor: What have we learned about turbulence?^{a)}
View Description Hide DescriptionTwenty years ago there was no experimental access to the velocity gradient tensor for turbulent flows. Without such access, knowledge of fundamental and defining properties of turbulence, such as vorticity dissipation, and strain rates and helicity, could not be studied in the laboratory. Although a few direct simulations at very low Reynolds numbers had been performed, most of these did not focus on properties of the small scales of turbulence defined by the velocity gradient tensor. In 1987 the results of the development and first successful use of a multisensor hotwire probe for simultaneous measurements of all the components of the velocity gradient tensor in a turbulent boundary layer were published by Balint et al. [Advances in Turbulence: Proceedings of the First European Turbulence Conference (SpringerVerlag, New York, 1987), p. 456]. That same year measurements of all but one of the terms in the velocity gradient tensor were carried out, although not simultaneously, in the selfpreserving region of a turbulent circular cylinder wake by Browne et al. [J. Fluid Mech.179, 307 (1987)], and the first direct numerical simulation (DNS) of a turbulent channel flow was successfully carried out and reported by Kim et al. [J. Fluid Mech.177, 133 (1987)], including statistics of the vorticity field. Also in that year a DNS of homogeneous shear flow by Rogers and Moin [J. Fluid Mech.176, 33 (1987)] was published in which the authors examined the structure of the vorticity field. Additionally, Ashurst et al. [Phys. Fluids30, 2343 (1987)] examined the alignment of the vorticity and strainrate fields using this homogeneous shear flow data as well as the DNS of isotropic turbulence of Kerr [J. Fluid Mech.153, 31 (1985)] who had initiated such studies. Furthermore, Metcalfe et al. [J. Fluid Mech.184, 207 (1987)] published results from their direct simulation of a temporally developing planar mixing layer in which they examined coherent vortical states resulting from secondary instabilities. Since then several experimentalists have used multisensor hotwire probes of increasing complexity in turbulent boundary layers, wakes, jets, mixing layers, and grid flows. Numerous computationalists have employed DNS in a wide variety of turbulent flows at ever increasing Reynolds numbers. Particle image velocimetry and other optical methods have been rapidly developed and advanced during these two decades which have provided other means of access to these fundamental properties of turbulence. This paper reviews highlights of these remarkable developments and points out some of the most important things we have learned about turbulence as a result.
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 LETTERS


Topological chaos and mixing in a threedimensional channel flow
View Description Hide DescriptionPassive mixing is investigated in a mathematical model of steady, threedimensional, laminar flow through a rectangular channel. Efficient stirring is achieved by imposing spatially periodic transverse boundary velocities that generate asymmetric, counterrotating rolls aligned with the channel axis. The flow is designed and analyzed using the concept of topological chaos, in which complexity is embedded in the flow through the motion of periodic orbits. A liddriven flow producing topological chaos is found to stir better than a related flow with solid inserts considered previously [M. D. Finn, S. M. Cox, and H. M. Byrne, Phys. Fluids15, L77 (2003)]. The results demonstrate that topological chaos and the Thurston–Nielsen classification theorem can provide insight into mixing enhancement in steady, threedimensional flows.
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 ARTICLES

 Interfacial Flows

Note on “Dynamics of inertia dominated binary drop collisions” [Phys. Fluids16, 3438 (2004)]
View Description Hide DescriptionThe analytical approach shown in the paper [I. V. Roisman, “Dynamics of inertia dominated binary drop collisions,” Phys. Fluids16, 3438 (Year: 2004)] has been found not to yield physically consistent results, for which the theoretically predicted initial kinetic energy of the radially expanding lamella is negative. Such theoretical predictions do not allow the ensuing computation of the flow field. The code for the numerical predictions of the evolution of the drop diameter includes a computational bug with which only a quarter of the dissipated energy is accounted for. Three possible sources of the error are identified: the assumed velocity field in the lamella, the shape of the lamella at the initial phase of drop deformation, and the energy balance formulation of the problem. A possible way for the improvement of the modeling of the dynamics of the initial stage of drop collision is proposed.

The splash of a solid sphere impacting on a liquid surface: Numerical simulation of the influence of wetting
View Description Hide DescriptionThe impact of a solid sphere on a liquid surface has challenged researchers for centuries and remains of interest today. Recently, Duez et al. [Nat. Phys.3, 180 (2007)] published experimental results of the splash generated when a solid sphere enters water. Interestingly, the microscopic properties of the solid surface control the nature of the macroscopic behavior of the splash. So by a change in the surface chemistry of the solid sphere, a big splash can be turned into an inconspicuous disappearance and vice versa. This problem was investigated by numerical simulations based on the Navier–Stokes equations coupled with the Cahn–Hilliard equations. This system allows us to simulate the motion of an airwater interface as a solid sphere impacts the liquid pond. The inclusion of the surface energies of the solid surface in the formulation gives a reasonably quantitative description of the dynamic wetting. Numerical results with different wetting properties and impact speed are presented and directly compared with the recent experimental results from Duez et al.

Lattice Boltzmann study of droplet motion inside a grooved channel
View Description Hide DescriptionA droplet moving inside a grooved channel is studied by using a new lattice Boltzmann model for multiphase flows with large density ratio. A constant body force is applied to drive the droplet.Flows under different surface tensions, driving forces, density ratios, wall wettabilities, and groove geometries are investigated. It is found that the drag on the droplet and the flow pattern are strongly affected by the wall wettability and topography when the system scale is small. The effects of the driving force on the droplet are investigated through comparison of two different ways of applying it. Besides, the density ratio is varied over a wide range to assess its effects in the present setup. Special attention is paid to grooved hydrophilic walls which tend to enhance the dropletwall contact. For such walls, two distinctive types of shape of the interface inside the groove are found and series of numerical investigations are carried out to find the critical wall contact angle, groove width and depth that determine which kind of shape the droplet assumes. Some typical cases are chosen for detailed analyses and compared to some other work. This study is expected to improve our understanding on the lotus effect and the physics of small scale flows near rough walls.

The stability of a bubble in a weakly viscous liquid subject to an acoustic traveling wave
View Description Hide DescriptionThe volume oscillations, translation, and axisymmetric deformation of a bubble in an acoustic traveling wave are considered. Assuming the bubble translation and deformation is small, but placing no restriction on the volume oscillations, a combination of the Rayleigh dissipation function and perturbation analysis is employed to account for the effects of viscosity in the absence of vorticity to third order in the small interaction terms. Contributions from the acoustic field are also determined to this order, while the free oscillation terms are drawn from a previously derived model correct to the same order of analysis. To permit the study of large amplitude acoustic forcing, appropriate compressibility terms are phenomenologically added to the volume pulsation equation. Stability maps of driving pressure versus driving frequency and driving pressure versus the equilibrium bubble radius are presented. A predominant number of results are for micronsized bubbles driven in the ultrasonic regime, but the behavior of larger bubbles driven at frequencies in the kilohertz range is also considered. In all cases, bubbles driven above the natural frequency of their respective volume oscillations are markedly more stable with regard to the acoustic driving amplitude, consistent with previous observations. Below these respective natural frequency values, the stability/instability fronts display a much more complex structure. Accounting for shape mode viscous damping causes a general increase in bubble stability, together with a reduction in the stability/instability front complexity. In the case of micronsized bubbles this stabilization is markedly more significant for bubbles driven above the natural frequency of the respective volume mode oscillations; for larger bubbles driven in the kilohertz range, the influence of shape mode damping is less significant.

Linear stability of a volatile liquid film flowing over a locally heated surface
View Description Hide DescriptionThe dynamics and linear stability of a volatile liquid film flowing over a locally heated surface are investigated. The temperature gradient at the leading edge of the heater induces a gradient in surface tension that leads to the formation of a pronounced capillary ridge. Lubrication theory is used to develop a model for the film evolution that contains three key dimensionless groups: a Marangoni parameter , an evaporation number , and a measure of the vapor pressure driving force for evaporation, which behaves as an inverse Biot number. The twodimensional, steady solutions for the local film thickness are computed as functions of these parameters. A linear stability analysis of these steady profiles with respect to perturbations in the spanwise direction reveals that the operator of the linearized system can have both a discrete and a continuous spectrum. The continuous spectrum exists for all values of the spanwise wave number and is always stable. The discrete spectrum, which corresponds to eigenfunctions localized around the ridge, appears for values of larger than a critical value for a finite band of wave numbers separated from zero. Above a second, larger critical value of , a portion of the discrete spectrum becomes unstable, corresponding to rivulet formation at the forward portion of the capillary ridge. For sufficiently large heat transfer at the free surface, due either to phase change or to convection, a second band of unstable discrete modes appears, which is associated with an oscillatory, thermocapillary instability above the heater. The critical Marangoni parameter above which instability develops, , has a nonmonotonic dependence on the steepness of the temperature increase at the heater, in contrast to the monotonic decrease for a nonvolatile film at vanishing Biot number. An energy analysis reveals that the dominant instability mechanism resulting from perturbations to the film thickness is either streamwise capillary flow or gravity for weakly volatile fluids and thermocapillary flow due to spanwise temperature gradients for more volatile fluids. The stability results are rather sensitive to the steepness of the temperature increase and heater width due to the nonlinear coupling of gravity, capillary pressure gradients, thermocapillary flow, and evaporation through the base states.
 Particulate, Multiphase, and Granular Flows

Initial rates of aggregation for dilute, granular flows of wet particles
View Description Hide DescriptionMolecular dynamics (MD) simulations are used to determine the agglomeration rates of wet grains (particles coated with a viscous, liquid layer) engaged in simple shear flow under dilute conditions. In this work, a closedform model derived from the elastohydrodynamic theory describes the normal restitution coefficient for binary collisions. Unlike previous MD studies, the particle deformation is not assumed to depend on a particle “overlap” (penetration), but instead depends on the formal coupling of Hertzian deformationtheory with lubricationtheory. The initial rate of doublet formation is studied as a function of systemproperties and the energy input to the system. In addition to the systemproperties, the distribution of relative velocities in the system is found to be a key factor influencing the initial rates of clustering. A theory based on estimating the collision frequency and the critical velocity below which no rebound is observed—due to viscous dissipation—is found to provide a good approximation of the initial aggregation rate. The rate of aggregation is found to increase with increasing number of particles in the system, increasing solid fraction, decreasing overall Stokes number, and decreasing compliance parameter.

Migration of rigid particles in nonBrownian viscous suspensions
View Description Hide DescriptionThere is an obvious need for obtaining a closed set of equations describing multiphase flows and the complexity of their configurations. Despite their “frozen” interfaces, suspensions of rigid particles are also concerned by this issue. The description of the relative motion between the two phases is still controversial, in particular, concerning the forces acting on the particles and involving their concentration gradient or a shear rate gradient. It is our purpose here (a) to develop a twofluid model especially designed for particles dispersed in a viscous liquid and (b) to close the model for rigid particles with the help of lowReynoldsnumber hydrodynamics. Besides the obvious role of gravity forces, the migration of particles relative to the fluid is shown to result from two different physical phenomena: (a) the inhomogeneity of the stress resulting from direct interparticle forces and (b) Ficklike terms in the hydrodynamic force acting on the particles.

Breakup of drops in a microfluidic T junction
View Description Hide DescriptionWe propose a mechanism of droplet breakup in a symmetric microfluidic T junction driven by pressure decrement in a narrow gap between the droplet and the channel wall. This mechanism works in a twodimensional setting where the capillary (Rayleigh–Plateau) instability of a cylindrical liquid thread, suggested earlier [D. Link, S. Anna, D. Weitz, and H. Stone, Phys. Rev. Lett.92, 054503 (Year: 2004)] as the cause of breakup, is not operative, but it is likely to be responsible for the breakup also in three dimensions. We derive a dependence of the critical droplet extension on the capillary number Ca by combining a simple geometric construction for the interface shape with lubrication analysis in a narrow gap where the surface tension competes with the viscous drag. The theory, formally valid for , shows a very good agreement with numerical results when it is extrapolated to moderate values of Ca.
 Laminar Flows

Behavior of cylindrical liquid jets evolving in a transverse acoustic field
View Description Hide DescriptionThis paper presents a theoretical and an experimental investigation of lowvelocity cylindrical liquid jets submitted to transverse planar acoustic waves. For this purpose, the behavior of a liquid jet traversing the section of a Kundt tube was examined. Experiments reported that the liquid jet could be either deviated from its trajectory or deformed as a succession of lobes oriented in space and whose length and width depend on the jet acoustic environment. Furthermore, for a sufficient acoustic velocity, the jet deformation increases in such proportion that a premature and vivid atomization mechanism disintegrates the liquid flow. Theoretical models are proposed to understand these behaviors. The first one calls out for acoustic radiation pressure to explain the jet deviation. The second one consists in a modal analysis of the vibrations of a jet when submitted to a transverse stationary acoustic field. As a first approach, a simplified twodimensional model is proposed. This model reports that a sudden exposition of the jet to an acoustic field triggers two jet eigenmodes. One of them induces jet deformations that were not experimentally observed. This part of the solution emerges due to theoretical deficiencies. However, the second mode reproduces the lobe formation and leads to atomization criteria in good agreement with the experimental results. The paper ends with an extension of the mathematical development in three dimensions in order to provide a basis to a more consistent model.

Estimation of volume flow in curved tubes based on analytical and computational analysis of axial velocity profiles
View Description Hide DescriptionTo monitor biomechanical parameters related to cardiovascular disease, it is necessary to perform correct volume flow estimations of blood flow in arteries based on local blood velocity measurements. In clinical practice, estimates of flow are currently made using a straighttube assumption, which may lead to inaccuracies since most arteries are curved. Therefore, this study will focus on the effect of curvature on the axial velocity profile for flow in a curved tube in order to find a new volume flow estimation method. The study is restricted to steady flow, enabling the use of analytical methods. First, analytical approximation methods for steady flow in curved tubes at low Dean numbers (Dn) and low curvature ratios are investigated. From the results a novel volume flow estimation method, the method, is derived. Simulations for curved tube flow in the physiological range ( and ) are performed with a computational fluid dynamics(CFD) model. The asymmetric axial velocity profiles of the analytical approximation methods are compared with the velocity profiles of the CFD model. Next, the method is validated and compared with the currently used Poiseuille method by using the CFD results as input. Comparison of the axial velocity profiles of the CFD model with the approximations derived by Topakoglu [J. Math. Mech.16, 1321 (1967)] and Siggers and Waters [Phys. Fluids17, 077102 (2005)] shows that the derived velocity profiles agree very well for and are fair for , and this result applies for , while Dean’s [Philos. Mag.5, 673 (1928)] approximation only coincides for . For higher Dean numbers , no analytical approximation method exists. In the position of the maximum axial velocity, a shift toward the inside of the curve is observed for low Dean numbers, while for high Dean numbers, the position of the maximum velocity is located at the outer curve. When the position of the maximum velocity of the axial velocity profile is given as a function of the Reynolds number, a “zeroshift point” is found at . At this point the shift in the maximum axial velocity to the outside of the curve, caused by the difference in axial pressure gradient, balances the shift to the inside of the curve, caused by the centrifugal forces (radial pressure gradient). Comparison of the volume flow estimation of the method with the Poiseuille method shows that for the Poiseuille method is sufficient, but for the method estimates the volume flow nearly three times better. For the maximum deviation from the exact flow is 4% for the method, while this is 12.7% for the Poiseuille method in the plane of symmetry. The axial velocity profile measured at a certain angle from the symmetry plane results in a maximum estimation error of 6.2% for and . The results indicate that the estimation of the volume flow through a curved tube from a given asymmetrical axial velocity profile is more precise with the method than the Poiseuille method, which is currently used in clinical practice.

Openflow mixing: Experimental evidence for strange eigenmodes
View Description Hide DescriptionWe investigate experimentally the mixingdynamics of a blob of dye in a channel flow with a finite stirring region undergoing chaotic advection. We study the homogenization of dye in two variants of an eggbeater stirring protocol that differ in the extent of their mixing region. In the first case, the mixing region is separated from the sidewalls of the channel, while in the second it extends to the walls. For the first case, we observe the onset of a permanent concentration pattern that repeats over time with decaying intensity. A quantitative analysis of the concentration field of dye confirms the convergence to a selfsimilar pattern, akin to the strange eigenmodes previously observed in closed flows. We model this phenomenon using an idealized map, where an analysis of the mixingdynamics explains the convergence to an eigenmode. In contrast, for the second case the presence of noslip walls and separation points on the frontier of the mixing region leads to nonselfsimilar mixingdynamics.
 Instability and Transition

Vortex formation in a cavity with oscillating walls
View Description Hide DescriptionThe vortex formation in a twodimensional Cartesian cavity, which their vertical walls move simultaneously with an oscillatory velocity and the horizontal walls are fixed pistons, is studied numerically. The governing equations were solved with a finite element method combined with an operator splitting scheme. We analyzed the behavior of vortical structures occurring inside a cavity with an aspect ratio of heighttowidth of 1.5 for three different displacement amplitudes of the vertical oscillatory walls (, 0.4, and 0.8) and Reynolds numbers based on the cavity width of 50, 500, and 1000. Two vortex formation mechanisms are identified: (a) the shear, oscillatory motion of the moving boundaries coupled with the fixed walls that provide a translational symmetrybreaking effect and (b) the sharp changes in the flowmotion when the flow meets the corners of the cavity. The vortex cores were identified using the Jeong–Hussain criterion and it is found that the area occupied by the cores decreases as the Reynolds number increases and increases as increases. All flows studied are cyclic symmetric and for low and Re values they are also symmetric with respect to the vertical axis dividing the cavity in two sides. In asymmetric flows, the unbalance between the vortices on each side of the midvertical line generates a vortex that occupies the central part of the cavity. The breakdown of the axial symmetry was studied for a fixed value of the oscillation amplitude, , taking Re as the bifurcation parameter. The results indicate that the symmetry is broken through a supercritical pitchfork bifurcation.

Modal and nonmodal growths of inviscid planar perturbations in shear flows with a free surface
View Description Hide DescriptionShear flows with a free surface possess diverse branches of modal instabilities. By approximating the mean flow with a piecewise linear profile, an understanding and classification of the instabilities can be achieved by studying the interaction of the edge waves that arise at the density discontinuity at the surface and the vorticitywaves that are supported at the mean vorticity gradient discontinuities in the interior. The various branches of instability are identified and their physical origin is clarified. The edge waves giving rise to the modal instabilities can also lead to a modest transient growth that extends into the regions of neutrality of the flow. However, when the continuous spectrum is excited substantial transient growth can arise and the optimal perturbations attain greater energy when compared with the energy of the fastest modal growing perturbation. These optimal perturbations utilize the continuous spectrum to excite at large amplitude the neutral or amplifying modes of the system.

Reshocks, rarefactions, and the generalized Layzer model for hydrodynamic instabilities
View Description Hide DescriptionWe report numerical simulations and analytic modeling of shock tubeexperiments on Rayleigh–Taylor and Richtmyer–Meshkov instabilities. We examine single interfaces of the type where the incident shock is initiated in and the transmitted shock proceeds into . Examples are He/air and air/He. In addition, we study finitethickness or doubleinterface configurations such as air//air gascurtain experiments. We first consider conventional shock tubes that have a “fixed” boundary: A solid endwall which reflects the transmitted shock and reshocks the interface(s). Then we focus on new experiments with a “free” boundary—a membrane disrupted mechanically or by the transmitted shock, sending back a rarefaction toward the interface(s). Complex acceleration histories are achieved, relevant for inertial confinement fusion implosions. We compare our simulation results with a generalized Layzer model for two fluids with timedependent densities and derive a new freezeout condition whereby accelerating and compressive forces cancel each other out. Except for the recently reported failures of the Layzer model, the generalized Layzer model and hydrocode simulations for reshocks and rarefactions agree well with each other and remain to be verified experimentally.

Simulation of the control of vortex breakdown in a closed cylinder using a small rotating disk
View Description Hide DescriptionThe enhancement or suppression of vortex breakdown in a closed cylinder caused by a small rotating disk embedded in the nonrotating endwall is simulated in this study. This paper shows that corotation or counterrotation of the control disk with respect to the driving lid is able to promote or suppress the “bubbletype” vortex breakdown. This is achieved using only a small fraction of the power required to drive the main lid. The simulations show that the vortex breakdown induced or suppressed by flow control displays similar characteristics near the breakdown region as produced by varying the flowReynolds number. These include nearaxis swirl, centerline axial velocity, and centerline pressure. The influence of the size of the control disk is also quantified.

Convective instability of flow in a symmetric channel with spatially periodic structures
View Description Hide DescriptionConvective instability of flow in a twodimensional symmetric channel with periodic suddenly expanded sections is investigated numerically under given inlet and outlet boundary conditions as a finitelength flow. A localized disturbance is added at the inlet of the channel and the subsequent spatiotemporal development of the disturbance is observed. The localized disturbance forms a similarly evolving wave packet while traveling downstream, which enables us to study the nature of convective instability. It is found that the wave packet consists of two intrinsic waves propagating with distinct phase velocities. The spatial structure and the phase velocity of these intrinsic waves are compared with the eigenmode of stability under two different periodic boundary conditions, one of which imposes the flow to have the same periodic length with that of the channel geometry and the other twice the periodic length together with a shiftandreflect symmetry. It is clarified that the two kinds of waves are well approximated by these two eigenmodes. Our results link a closed flow under periodic boundary conditions to an open flow in a finitelength channel with periodic structures, which enables us to estimate the critical Reynolds number of convective instability and the traveling velocity of the wave packet.

Wall heat transfer effects on Klebanoff modes and Tollmien–Schlichting waves in a compressible boundary layer
View Description Hide DescriptionThe influence of wall heat transfer on fluctuations generated by freestream vortical disturbances in a compressible laminar boundary layer is investigated. These disturbances are thermal Klebanoff modes, namely lowfrequency, streamwiseelongated laminar streaks of velocity and temperature, and oblique Tollmien–Schlichting waves, induced by a leadingedge adjustment receptivity mechanism. The flow is governed by the linearized unsteady boundaryregion equations, which properly account for the nonparallel and spanwise diffusion effects, and for the continuous forcing of the freestream convected gusts. Wall cooling stabilizes the laminar streaks when their spanwise wavelength is much larger than the boundarylayer thickness. For these conditions, the disturbances confine themselves in the outer edge layer further downstream, where the compressibility effects are marginal. Klebanoff modes for which the spanwise diffusion is comparable with the wallnormal diffusion possess an asymptotic solution similar to the incompressible case, and are stabilized by wall heating. The unstable waves, which appear in highMachnumber subsonic and supersonic conditions, are stabilized by wall cooling and destabilized by wall heating. Removing heat from the surface significantly shifts downstream the starting location of instability, while the streamwise wavelength and the growth rate are less affected by the wall heat flux. Perturbation methods, such as the WKBJ technique and the tripledeck theory, are used effectively to validate the numerical results and to explain the flow physics.
 Turbulent Flows

Modeling the effect of upstream temperature fluctuations on shock/homogeneous turbulence interaction
View Description Hide DescriptionReynoldsaveraged Navier–Stokes (RANS) equations can yield significant error when applied to the interaction of turbulence with shock waves. This is often due to the fact that RANS turbulencemodels do not account for the underlying physical phenomena correctly. For example, in the presence of appreciable temperature fluctuations in the upstream flow,turbulence amplification across a shock is significantly affected by the magnitude of the temperature fluctuations and the sign of the upstream velocitytemperature correlation [Mahesh et al., J. Fluid Mech.334, 353 (1997)]. Standard twoequation models with compressibility correction do not reproduce this effect. We use the interaction of homogeneous isotropic turbulence with a normal shock to suggest improvements to the model. The approach is similar to that presented in an earlier work [Sinha et al., Phys. Fluids15, 2290 (2003)]. Linear inviscid analysis is used to study the effect of upstream temperature fluctuations on the evolution of turbulent kinetic energy across the shock. The dominant mechanisms contributing to the amplification are identified and then modeled in a physically consistent way. The dissipation rate equation is also altered based on linear analysis results. The modifications yield significant improvement over existing twoequation models and the new model predictions are found to match linear theory and direct numerical simulation data well.