Modelling of turbulent energy flux in canonical shock-turbulence

interaction

Russell Quadros and Krishnendu Sinha

Indian Institute of Technology Bombay, Mumbai, India 400076

Abstract

High-speed turbulent flows often encounter high heat loads due to the presence of shockwaves. The turbulent energy flux correlation in the mean energy conservation equationis a key unclosed term that determines the heat transfer rate. In this work, we employexisting turbulence models to predict the turbulent energy flux in canonical shock-turbulenceinteraction. The shortcomings of these models are highlighted, and a new heat-flux limitermodel is proposed with the aid of linear theory results. We also write the transport equationfor the turbulent energy flux across a shock wave and use it to develop a physics-based modelfor the same. It is found to predict the peak energy flux at the shock wave and its variation inthe acoustic-adjustment region behind the shock. Numerical error incurred while solving themodel equations at a shock wave are analyzed and a numerically robust model is obtainedby eliminating the nonconservative source terms. The model predictions are compared withavailable direct numerical simulation data and a good match is obtained for a range of Machnumbers.

Keywords: High-speed flows, Heat transfer, Linear interaction analysis, RANS modelling

1. Introduction

High-speed flows in aerospace applica-tions have shock waves interacting withboundary layers in different parts of thevehicle surface and in engine compo-nents. Such shock/boundary-layer interac-tions (SBLI) are often marked by high local-ized surface pressure and heat transfer rates[1, 2, 3]. Predicting the heating loads is es-pecially important in supersonic and hyper-sonic applications with turbulent boundarylayers. Majority of the existing turbulencemodels for heat flux prediction yield accept-able results in boundary-layer flows [4], but

their predictive capability is severely limitedin shock-dominated flows [5].

An important unclosed term in the meanenergy conservation equation, which gov-erns the heat transfer rate, is the turbu-

lent energy flux vector �u��j e

��. Here, u��j rep-

resents the velocity fluctuation in the jth

direction, e�� represents the internal energyfluctuation and tilde represents Favre aver-aging. Conventionally, this term is mod-elled as a product of turbulent conductivityand the gradient of mean temperature. Tur-bulent conductivity is related to the eddyviscosity µT via a turbulent Prandtl number

Preprint submitted to International Journal of Heat and Fluid Flow September 8, 2016

This article is publishded in IJHFF Journal and can be downloaded from this link: http://dx.doi.org/10.1016/j.ijheatfluidflow.2016.07.006

U1Uniform mean flow U2

y

Distorted shock

x

Incoming turbulence

Figure 1: Schematic showing a shock wave distortedupon interaction with turbulent fluctuations.

PrT . A constant value of PrT = 0.89 givessatisfactory results in flat plate boundarylayers and is often used in SBLI configura-tions. This modelling approach however sig-nificantly overpredicts the actual wall heattransfer rate in SBLI [5].

An alternate model based on variable PrTapproach is proposed by Xiao et al. [6]. Itsolves additional differential equations forenthalpy variance and its dissipation rate,and employs these two quantities in the for-mulation of turbulent heat flux vector. Inshock-dominated flows, this approach leadsto improved wall heat flux predictions, yetit overpredicts experimental data. Som-mer et al. [7] also develop a variable turbu-lent Prandtl number (PrT ) model for high-speed compressible turbulent boundary lay-ers with adiabatic and isothermal wall con-ditions. Other efforts in this direction areby Goldberg et al. [8], Aouissi et al. [9] andAbe and Suga [10]. Another notable workis by Bowersox [11], which proposes an al-gebraic model for the turbulent energy fluxin supersonic flows, but is limited to zero-pressure-gradient boundary layers withoutshock waves.

In a recent work, Quadros et al. [12]present a detailed study of the turbulent

energy flux at a shock wave. They in-vestigate the physical processes that gov-ern the generation of the energy flux cor-relation in a canonical shock-turbulence in-teraction (STI). This model problem con-sists of homogeneous isotropic turbulence,which is purely vortical in nature, beingcarried by a one-dimensional uniform meanflow passing through a nominally normalshock wave. The turbulence is amplifiedby the shock, and the shock in turn getsdistorted. Schematic of this problem isshown in figure 1. This is possibly the sim-plest configuration that isolates the effectof shock on turbulence without other ef-fects, such as boundary-layer gradient, flowseparation and streamline curvature. Inspite of its geometrical simplicity, STI dis-plays a range of physical effects, such as,turbulence anisotropy, generation of acous-tic waves, baroclinic torques, and unsteadyshock oscillations.The work presented in Quadros et al.

[12] relies primarily on linear interactionanalysis (LIA), a theoretical approach toanalyse canonical STI. The analysis in-volves solving the fundamental interactionof a single two-dimensional plane wavewith a shock, which generates downstreamdisturbances that can be characterised interms of Kovasznay modes of vorticity, en-tropy and acoustic [13]. Integration overa specified upstream turbulence spectrumyields three-dimensional turbulence statis-tics downstream of the shock. The LIAresults obtained for the energy flux arecompared with direct numerical simula-tion (DNS) data available from Larssonet al. [14]. The results show a rapid non-monotonic variation behind the shock wave,and a peak positive value of the correlationin the near-field acoustic adjustment region.Quadros et al. [12] also compute the bud-

2

get of the energy flux transport equation us-ing the DNS data. It is found that the rapidpost-shock variation of the turbulent energyflux is governed by inviscid-linear mecha-nisms, and there is negligible effect of theviscous and non-linear terms to the overallbudget. The linear inviscid theory is ableto predict the dominant mechanisms andthus can be effectively used for developingadvanced physics-based models for turbu-lent energy flux generated at shock waves.This is similar to the LIA-based modellingof the turbulent kinetic energy amplificationat shock waves [15, 16], and the resultingshock-unsteadiness model has shown sig-nificant improvements in SBLI predictions[17, 18, 19].In another recent work, Quadros et al.

[20] study the correlation coefficient RuT be-tween streamwise velocity and temperaturefluctuations in canonical shock-turbulenceinteraction. RuT is a key link in mod-elling of turbulent energy flux. As perthe strong Reynolds analogy, the veloc-ity and temperature fluctuations are anti-correlated, i.e. RuT = −1. This assump-tion corresponds to a value of turbulentPrandtl number PrT = 1, which is closeto the value used in practice. In the pres-ence of shock waves, Morkovin’s hypothesisis not valid [21], and RuT can vary signifi-cantly. Quadros et al. [20] present the vari-ation of RuT with Mach number, turbulentMach number and Reynolds number, as pre-dicted using linear inviscid theory and com-pare to the data from DNS. Different typesof incoming disturbances – purely vortical,vortical/entropy and vortical/acoustic – arealso considered.Quadros et al. [20] decompose the

velocity-temperature correlation coefficientin terms of the three Kovasznay modes ofvorticity, entropy and acoustic. The con-

tributions from the individual Kovasznaymodes are quantified, and compared withavailable DNS data. At low Mach numbers,it is found that the peak post-shock RuT isdetermined by the acoustic mode, which iscorrectly predicted by the linear theory. Athigh Mach numbers, the post-shock RuT isdetermined primarily by the vorticity andentropy modes, which are strongly affectedby nonlinear and viscous effects, and thusless well predicted by linear theory.In this paper, we investigate different

modelling strategies to predict the tur-bulent energy flux generated by shock-homogeneous turbulence interaction. It isbased on the physical understanding of howthe turbulent energy flux evolves across ashock wave, at varying Mach and Reynoldsnumbers, as per our recent works summa-rized in section 2. We consider existing tur-bulence models, including conventional gra-dient diffusion and realizable models (sec-tion 3). We also develop new models basedon linear inviscid theory (section 4) andstudy their numerical behaviour at shockwaves (section 5).

2. Turbulent energy flux in shock-turbulence interaction

Quadros et al. [12] studied the generationof turbulent energy flux in canonical STIproblem using LIA, which models the up-stream turbulence as a collection of planarwaves. Each of these waves is consideredto independently interact with the shock.The governing equations downstream of theshock are linearized Euler equations andthe jump in fluctuations across the shockis given by linearised inviscid Rankine-Hugoniot conditions.A set of linear algebraic equations are ob-

tained by substituting the planar waveforms

3

M1 Mt Reλ Rkk/(2U21 ) κ◦L� k0 × 102 �0 × 103 k1 × 102 �1 × 103 κ◦δs

1.28 0.22 38 7.3× 10−3 3.9 2.98 1.17 2.42 0.92 1.751.50 0.22 38 5.3× 10−3 3.9 2.88 1.12 2.42 0.92 1.301.87 0.22 39 6.9× 10−3 3.8 2.78 1.08 2.42 0.92 1.182.50 0.22 40 4.0× 10−3 4.3 2.69 1.03 2.42 0.92 0.863.50 0.22 40 2.1× 10−3 4.3 2.61 1.00 2.42 0.92 0.454.70 0.23 42 1.2× 10−3 4.3 2.56 0.97 2.42 0.92 0.436.00 0.23 42 0.7× 10−3 4.4 2.53 0.96 2.42 0.92 0.36

Table 1: DNS cases from Larsson et al. [14] with mean Mach number, turbulent Mach number and Reynoldsnumber listed at a location just upstream of the shock. The peak wavenumber κ◦ is used to normalize thedissipation length scale L� and the DNS mean shock thickness δs. Also given are the non-dimensional valuesof turbulence kinetic energy k and dissipation rate � at the inlet station (subscript 0) and just before theshock (subscript 1).

into the governing equations. Solving theseequations yields the disturbance field down-stream of the shock for a given upstreamvortical wave. The downstream turbulentfield for a full three-dimensional upstreamturbulence is obtained by integrating thetwo-dimensional planar wave results over aspecified energy spectrum. Details of thisanalysis can be found in Mahesh et al. [22].

DNS of canonical STI was carried outby Larsson et al. [14] for vortical turbu-lence passing through a normal shock. Ta-ble 1 lists the DNS cases used in the cur-rent study. The Mach numbers range fromlow supersonic to hypersonic values. Turbu-lent Mach number Mt and Reynolds num-ber based on Taylor scale Reλ for each ofthe cases are also listed. Here, turbulentMach number is defined as Mt =

√Rkk/�a,

where Rkk represents twice the turbulencekinetic energy, and �a represents the Favre-averaged speed of sound. Reλ is given byρ�

Rkk/3λ/µ, where λ is the Taylor scale

and µ is the average dynamic viscosity.Figure 2(a) compares the streamwise vari-

ation of the turbulent energy flux ob-tained from DNS and LIA for M1 =1.50 (reproduced from Quadros et al. [12]).Note that conventional Reynolds averag-ing/fluctuation is used instead of its Favrecounterpart, as negligible difference is ob-served upon comparison using the DNS dataset. The shock is centered at x = 0 withthe unsteady shock movement region high-lighted using two vertical lines. We normal-ize the streamwise distance with dissipationlength scale L�, calculated just upstream ofthe shock as

L� = (Rkk/2)3/2/�, (1)

where � is the dissipation rate of turbu-lence kinetic energy, and the values of κ◦L�

for DNS are listed in table 1. Since vis-cous dissipation is absent in the linear the-ory, the equivalent dissipation length scaleis obtained by using the most energetic

4

x/Lε

0 2 4

0

0.1

0.2

0.3

LIA

Increasing Re

(a)

M1

1 3 5 7-0.5

0

0.5

1

1.5

2

(b)

Figure 2: Turbulent energy flux generated in shock-turbulence interaction. (a) Streamwise variation ofturbulent energy flux for M1 = 1.5 from LIA (lines) and DNS (line-symbols) with Mt = 0.15. Effect ofReynolds number is also shown with Reλ = 40 (circle) and Reλ = 75 (square). x = 0 represents the centerof the mean shock thickness, and the vertical lines near x = 0 represent the mean shock thickness region. (b)Variation of peak u�

2e�2 with upstream Mach number. The DNS data corresponds toMt = 0.22 and Reλ = 40.

The velocity fluctuations are normalized by the upstream mean velocity and the energy fluctuations arenormalized by the product of the downstream mean temperature and the specific gas constant R. Thecorrelation is further normalized by upstream turbulence kinetic energy.

wavenumber (κ◦) in the expression κ◦L� ≈3− 4 obtained from the DNS study [14].

Upstream of the shock, DNS shows a neg-ligible value of turbulent energy flux, andin the region of the unsteady shock wave, itpredicts large negative values of the corre-lation. Just downstream of the shock, bothLIA and DNS predict a peak positive en-ergy flux. Further downstream, DNS showsa steep decrease in the energy flux valuesto negligible levels, whereas LIA predictsa constant far-field value. With increasingReynolds number, the DNS data tends toapproach the LIA prediction, which is rep-resentative of Re → ∞ limit. The peakturbulent energy flux obtained from DNSat two Reynolds numbers are comparablein magnitude, and their location are closeto each other, when the streamwise coordi-nate is normalized by the dissipation length

scale. This is not the case when Kolmogorovscale is used to non-dimensionalize the dis-tance from the shock wave.

The peak value of turbulent energyflux u�

2e�2 (subscript 2 indicates the shock-

downstream values) as predicted by bothDNS and LIA for varying Mach numbersis shown in figure 2(b), reproduced fromQuadros et al. [12]. Also shown is thefar-field asymptotic value of LIA, and athigh Mach numbers, it is almost equal tothe LIA peak energy flux. The peak en-ergy flux as per theory matches well withDNS for low Mach numbers upto M1 < 2,with the theory overpredicting the values athigher shock strengths. The LIA predictionreaches an asymptotic value of 1.32 at highMach numbers, while DNS shows a limit-ing value less than 1. The DNS data isfound to approach the LIA predictions with

5

M1

1 2 3 4 5 6-400

-300

-200

-100

0

Figure 3: Peak turbulent energy flux for varyingupstream Mach numbers as predicted by conven-tional model with the shock-thickness taken fromthe DNS data. The results correspond to turbu-lent Mach number Mt = 0.22 just before the shock.Normalisation as described in figure 2.

decreasing turbulent Mach number (lowernon-linear effects); see Quadros et al. [12]for further details. Overall, a good quali-tative match is observed between LIA andDNS indicating that the key physical mech-anisms governing the energy flux transportare captured by the theory.

3. Existing models for turbulent en-ergy flux

3.1. Conventional Modelling

The turbulent energy flux is convention-ally modelled using the gradient diffusionhypothesis as

u�e� = −κT

ρ

∂ �T∂x

, (2)

where ρ and �T are mean density and tem-perature. Thermal conductivity κT is givenby κT = µT cv/PrT , where cv is the specificheat at constant volume and PrT represents

the turbulent Prandtl number having a con-stant value of 0.89. Eddy viscosity is givenby µT = c◦µρk

2/�, where c◦µ = 0.09, k is tur-bulence kinetic energy (TKE). In order tocompute for the turbulent energy flux in thegiven model problem, the mean flow vari-ables are prescribed as hyperbolic tangentprofiles across the shock, with the shockthickness obtained from the DNS data. Thevalues of k and � used in the expressionfor eddy viscosity are obtained by solvingthe corresponding differential equations [15]using a fourth order Runge-Kutta (RK4)method, with the inlet boundary conditionspecified from the DNS data. The resultingvalues for k and � match well with DNS asreported in earlier works [15, 16].

The value of the energy flux obtained us-ing (2) is zero in the region upstream anddownstream of the shock due to uniformmean flow temperature on either side of thenormal shock. However, the energy flux as-sumes a peak negative value in the regionof the shock, and the peak values obtainedfor a range of upstream Mach number areshown in figure 3. In the limit of M1 → 1,the value of the energy flux predicted by themodel reduces to zero, and with increasingMach number, the magnitude of the nega-tive peak rises to a value that is almost twoorders of magnitude higher than the post-shock DNS predictions. Also, in a CFDframework, this formulation yields a grid-dependent value of the energy flux in the re-gion of the shock From (2), u�e� ∼ Δ�T/δs ∼Δ�T/Δx, where Δ�T is the jump in the meantemperature across the shock, δs is the meanshock thickness and Δx is the local gridresolution. Increasing the grid point den-sity (decreasing Δx) therefore increases themagnitude of the negative peak.

6

M1

2 3 4 5 6 7 8-1.5

-1

-0.5

0

Realizable

Model

M1

1 1.005 1.01-0.4

-0.3

-0.2

-0.1

0

Figure 4: Variation of peak turbulent energy fluxwith upstream Mach number as predicted by re-alizable model with shock thickness taken fromthe DNS data. The peak value is independent ofthe shock thickness, which is indicative of a grid-independent solution in a CFD simulation. Nor-malisation as described in figure 2.

3.2. Realizable Model

In canonical shock-turbulence interac-tion, the Reynolds stress as predicted by theconventional model is given by [15]

ρu�u� = −4

3µT

∂�u∂x

+2

3ρk. (3)

This modelling approach leads to unphys-ically high values of TKE in the regionof shock due to its proportionality withthe mean velocity gradient. The realiz-ability correction limits the value of eddyviscosity in the region of shock throughthe form cµ = min(coµ,

�coµ/s) [23], where

s = S/(�/k), S =�

(2SijSji − (2/3)S2kk)

and Sij = (∂�ui/∂xj + ∂�uj/∂xi)/2. In theregion of high gradients such as shock cµ =(�

coµ�)/(Sk) and therefore

µT =

�3coµ2

ρk��∂�u∂x

�� . (4)

Thus the expression for energy flux givenby (2) reduces to

u�e� = −�

3c0µ2

kcv

PrT��∂�u∂x

��∂ �T∂x

, (5)

in the shock region.The value of TKE as well as the mean

variables required for the above formulationare obtained as described in the previoussection. Similar to the conventional model,the realizable model yields a zero value ofenergy flux outside the region of shock.However, inside the shock region, a peaknegative value is obtained as per (5). Thispeak energy flux value for varying Machnumber is plotted in figure 4. For almostall Mach numbers, the realizable model pre-dicts a negative energy flux in the shock re-gion, but the magnitude is restricted dueto the realizability constraint, and is of thesame order as the post-shock DNS value.Contrary to the eddy viscosity formulation,the realizability limiter yields a peak valueof energy flux which is independent of theshock thickness assumed. This is becauseu�e� ∼ (Δ�T/δs)/(Δ�u/δs) ∼ Δ�T/Δ�u, assum-ing k to be finite, as predicted in Sinha et al.[15]. Thus, in a real CFD simulation, thepeak value in the shock region will be in-sensitive to grid refinement. In the limitof M1 → 1, the realizable model switchesto the conventional model formulation andpredicts a value of zero energy flux as shownin the inset figure. In the limit of M1 → ∞,the model saturates to a negative limitingvalue, a trend similar to the DNS data, butopposite in sign.

4. New physics-based models

4.1. Algebraic heat flux limiter model

We use linear interaction analysis resultsto propose an algebraic model for turbulent

7

energy flux at the shock as

u�e� = βkcv��∂�u∂x

��∂ �T∂x

. (6)

This form is similar to the realizability con-straint shown in (4), with β as an unknownmodelling parameter. From the mean en-ergy conservation equation across the shock,we can write

�u����∂�u∂x

���� = cp∂ �T∂x

, (7)

where cp is the specific heat at constantpressure. Substituting the above equationin (6) yields

u�e� =βk�uγ

, (8)

where the turbulent energy correlation isseen to be proportional to turbulent kineticenergy. This is physically consistent withthe linear theory results that show all cor-relations generated across the shock to bedirectly proportional to the upstream TKE.Further, the normalised expression for en-

ergy flux can be written as

u�e�|Norm. =u�e�

R�u1�T2

k1�u21

, (9)

where R is the specific gas constant. Theenergy flux is normalized in this mannerthroughout this work, and the subscriptNorm. is dropped for convenience. Using(8) with the upstream values of k and U ,along with (9), simplifies the above expres-sion to

u�e� = βM2

1

Tr

, (10)

where Tr is the mean temperature ratioacross the shock. In the high Mach num-ber regime, this ratio is proportional to M2

1 ,

M1

1 2 3 4 5 6 70

1

2

LIADNS

Algebraic model

Figure 5: Variation of u�e� with upstream Machnumber as per new algebraic formulation. Also seenare the DNS values of energy flux extrapolated tothe shock center and the far-field LIA results. Nor-malisation as described in figure 2.

and the above expression obtains a limitingvalue

u�e� = β(γ + 1)2

2γ(γ − 1). (11)

A similar trend is observed in DNS and LIA,where the peak turbulent energy flux satu-rates at high Mach numbers. In order tofind the value of the modelling parameterβ, we equate (11) to the LIA far-field lim-iting value of 1.4. This gives a value ofβ = 0.27 for γ = 1.4. However, in the limitof M1 → 1, the energy flux attains a valueequal to 0.27, which is physically incorrect.On the contrary, the energy flux predictedby the conventional model in the M1 → 1limit is zero (as u�e� is proportional to themean temperature gradient), which is con-sistent with the DNS and LIA predictions.We therefore propose a low Mach numbercorrection of the form

β� = (1− e1−M1)β, (12)

which yields β� = 0 as M1 → 1 and β� → βas M1 → ∞.

8

Figure 5 shows the model predictions ofthe peak turbulent energy flux for a rangeof upstream Mach numbers. Also shown inthe figure are the downstream DNS valuesof energy flux extrapolated to the shock cen-ter and the far-field LIA values. The modelmatches well with DNS for M1 < 2, and forhigher Mach numbers, it is close to LIA andoverpredicts the DNS values.

4.2. Differential-equation based model

In this section, we propose a differential-equation based model to predict the turbu-lent energy flux in STI. The model devel-opment is based on the linear theory ap-proximations, and it builds on the fact thatthe peak turbulent energy flux behind theshock is reproduced well by LIA. The modelis developed in two parts. The first partattempts to predict the peak turbulent en-ergy flux generated behind the shock, andthe second part reproduces the decay of theenergy flux further downstream. The modelis developed in conjunction with the modelequations for TKE and its dissipation rateproposed by Sinha et al. [15]. The closurecoefficients are obtained from the LIA re-sults, as in the earlier works [15, 24, 16], andthe model predictions are compared withavailable DNS data [14]. The equation forthe conservation of total enthalpy across theshock wave can be written in terms of theshock-normal velocity component un as

∂he

∂x+ un

∂un

∂x= 0, (13)

where he is the enthalpy. For an unsteadydistorted shock wave of the form presentedin figure 1, un � u+ u� − ξt, which assumessmall deviation of the shock wave from itsmean location. Here, ξ represents the shockdisplacement from its mean position, andξt represents the local streamwise velocity

M1

1 2 3 4 5 6 7-1

-0.5

0

0.5

1

1.5

2

Second Term

First Term

Figure 6: Variation of u�2e

�2 with upstream Mach

number as predicted by (16) along with the indi-vidual contributions of the source terms. Normali-sation as described in figure 2.

of the shock wave; see Sinha [16] for fur-ther details. Linearising the above equationin terms of fluctuations in enthalpy h� andstreamwise velocity u�, we get

∂h�

∂x+ u

∂u�

∂x+ u�∂u

∂x− ξt

∂u

∂x= 0. (14)

Taking a moment with u� and Reynolds av-eraging yields a differential equation for theenergy flux correlation.

γ∂

∂xu�e� = −u�2∂u

∂x+u�ξt

∂u

∂x−u

∂

∂x

u�2

2+h�∂u

�

∂x,

(15)where the enthalpy fluctuations are replacedby the internal energy fluctuations via h� =γe�. The first term on the RHS repre-sents the generation of turbulent energyflux due to mean compression in the shockwave, and the second term brings in theshock-unsteadiness effect. The change inthe streamwise Reynolds stress across theshock contributes to the turbulent energyflux via the third term on the RHS. Thelast term is a correlation of the enthalpy

9

fluctuations with the change in the stream-wise velocity fluctuations across the shock.The first two source terms are analo-

gous to the production and the shock-unsteadiness damping terms in the TKEequation presented in Sinha et al. [15]. Wefollow their modelling approach to writeu�ξt = b1u�2, which is based on the assump-tion that the unsteady shock oscillationsare caused by the incoming velocity fluc-tuations. The streamwise Reynolds stressis then closed in terms of the turbulencekinetic energy by considering its isotropicform i.e., u�2 = 2k/3 [15]. The final modelequation for the turbulent energy flux thustakes the form

∂

∂x

�u�e�

�= −2(1− b1)

3γk∂u

∂x− 1

3γu∂k

∂x,

(16)where b1 = 0.4 + 0.6 exp(2(1 − M1)). Thelast term is dropped from the equation forwant of an adequate closure approximation.The value of turbulent energy flux is ob-

tained by numerically integrating (16) overprescribed hyperbolic tangent shock pro-files of the mean variables. Figure 6 showsthe value of turbulent energy flux obtainedacross the shock for varying upstream Machnumbers. The correlation takes a value ofzero as M1 → 1, which is physically con-sistent with negligible turbulent energy fluxacross very weak shocks. With increasein upstream Mach number, the energy fluxgradually rises to saturate in the M1 → ∞limit. The contribution of both the termson the RHS of (16) are also shown in thefigure. While the first model term in theabove equation generates a positive turbu-lent energy flux across the shock wave, thesecond term has a negative contribution tothe energy flux correlation. The magnitudeof the first term is significantly higher thanthe second for all upstream Mach numbers,

therefore yielding a positive value of the en-ergy flux.Now consider the differential equation

governing the jump in TKE at the shockgiven by the shock-unsteadiness model [15]

ρ�u∂k∂x

= −2

3ρk

∂�u∂x

(1− b1). (17)

Also, from the mean energy conservationequation across the shock, we can write

�u����∂�u∂x

���� = cp∂ �T∂x

. (18)

Using the above two equations, (16) can bewritten as

∂

∂x

�u�e�

�= cue

�k

�u

�cp∂ �T∂x

, (19)

where cue = 4(1− b1)/(9γ) and �u � u. Thisequation is compact with only one mod-elled term governing the value of energy fluxacross the shock. Moreover, similar to theconventional form discussed earlier, the en-ergy flux is modelled in terms of the gra-dient in the mean temperature. Figure 7compares the energy flux values as predictedby (19) for varying upstream Mach numberswith the far-field LIA results and the down-stream DNS values of energy flux extrapo-lated to the shock center. The model pre-diction attains a value of zero as M1 → 1and saturates to a positive limiting valueat high Mach numbers similar to the DNSand LIA results. A good match is observedwith the LIA results, and the model over-predicts in comparison with the DNS datafor higher upstream Mach numbers. In or-der to model the decay in the turbulent en-ergy flux behind the shock, we look at themechanisms responsible for its post-shockevolution. Quadros et al. [12] study the

10

x/Lε

0 0.5 1 1.5 2

0

0.15

0.3

M1=1.50

NC

DNS

RK4

(a)

x/Lε

0 0.5 1 1.5 2

0

0.2

0.4

M1=1.87

DNS

NC

RK4

(b)

x/Lε

-0.5 0 0.5 1 1.5 2

0

0.2

0.4

0.6 M1= 2.50NC

RK4

DNS

(c)

x/Lε

-0.5 0 0.5 1 1.5 2

0

0.4

0.8

1.2 M1= 3.50

NC

DNS

RK4

(d)

Figure 8: Streamwise variation of u�e� as computed using Runge-Kutta (RK4) method (line) and the Lax-Friedrich scheme (dash-dot) for the nonconservative (NC) form of transport equation, compared with theDNS data (symbols) for four upstream Mach numbers. The vertical lines near x/L� = 0 represent the meanshock thickness. Normalisation as described in figure 2.

budget of u�e� transport equation in the lin-ear inviscid framework, which yields

ρ �u ∂

∂xu�e� = − e�

∂p�

∂x� �� �Press. energy

− pu�∂u�j

∂xj

,

� �� �pressure-dilatation

(20)i.e., the post-shock turbulent energy fluxis determined by the pressure-energy andpressure-dilatation mechanisms. In the re-gion behind the shock (0 < x/L� < 1), theseprocesses are dominated by the acoustic

mode, for which the thermodynamic fluctu-ations can be assumed to be approximatelyisentropic. Therefore, the pressure-energyterm can be written as [12]

−e�∂p�

∂x= − 1

γρ

∂

∂x

�p�2

2

�,

which corresponds to the streamwise vari-ation of pressure variance. The pressure-dilatation term in (20) is proportional tothe mean post-shock pressure, and the

11

M1

1 3 5 7

0

1

2

LIA

DNS

RK4

Figure 7: Variation of u�2e

�2 with upstream Mach

number as predicted by (19) using the Runge-Kutta(RK4) integration method. Also shown are the cor-responding DNS values extrapolated to shock cen-ter and far-field LIA results. Normalisation as de-scribed in figure 2.

u�(∂u�j/∂xj) correlation can be expanded as

u�∂u�j

∂xj

=∂

∂x

u�2

2+ u�∂v

�

∂y+ u�∂w

�

∂x.

The first term on the RHS represents thebuildup of streamwise Reynolds stress asa result of the acoustic decay, and canbe argued to be positive in the acoustic-adjustment region behind the shock.

Data from DNS and LIA [20] show thatthe pressure variance p�2 decays exponen-tially behind the shock, and the same canbe expected for u�e�. This decay can bemodelled by adding a term proportional to−u�e�/L to the RHS of the u�e� transportequation. Here, the characteristic decaylength scale L is taken as the dissipationlength scale L�, which is representative ofthe large acoustic scales in the turbulencefield. The full model equation for the en-

ergy flux thus takes the following form

∂

∂x

�u�e�

�= cue

�k

�u

�cp∂ �T∂x

− c2u�e�

L�

, (21)

where c2 = 0.3+3 exp(1−M1) is a modellingparameter based on the DNS data.

Figure 8 shows the streamwise variationof the turbulent energy flux for four up-stream Mach numbers, obtained by numeri-cally integrating (21) using the fourth-orderRunge-Kutta (RK4) method. The shock isspecified as a hyperbolic tangent profile ofthe mean flow variables; it is centered atx = 0, with the thickness obtained from theDNS data (shown using two vertical lines).The turbulent energy flux in the incomingflow is set to zero for the purely vortical tur-bulence upstream of the shock wave.

The model predicts a peak positive valueof the correlation across the shock, and weobserve a good match between the modelprediction and the DNS data for M1 = 1.5and 1.87. For the Mach 2.5 case, the modeloverpredicts the peak u�e� observed in DNS,but matches the post-shock decay rate upto x = L�. At the highest Mach numberinteraction considered here (M1 = 3.5), themodel overpredicts the peak value as well asthe decay rate behind the shock for x < L�.Further downstream, we see a systematicunder-prediction for all Mach numbers, andit is most prominent for the Mach 2.5 case.This can be attributed to the fact that themajority of the pressure fluctuations decaywithin x � L�, and further decay of u�e�

is possibly governed by viscous dissipation.The current model, without any viscous ef-fects, is unable to reproduce u�e� outside theacoustic decay region behind the shock.

12

5. Numerical simulation

The model predictions presented aboveare obtained by numerically integrating themodel equations for specified mean flow pro-files across a shock wave. The integrationstep size is kept small so that the resultsare devoid of numerical error. On the otherhand, in computational fluid dynamic simu-lations presented below, the mean flow vari-ables at a shock wave are computed as partof the solution, and are prone to numericalerror. In this section, we assess the effect ofthe numerical error on the turbulent energyflux prediction and device ways to eliminatethem.

5.1. Numerical methodology

We solve the one-dimensional Reynolds-averaged Navier-Stokes equations for the

mean flow, and the shock-unsteadiness k-� model from Sinha et al. [15] is used forturbulence closure. The diffusive fluxes areneglected because their contribution is ex-pected to be small in the current shock-turbulence interaction [25]. The govern-ing equations are cast in a non-dimensionalform, where the upstream mean speed ofsound and mean density are taken as thecharacteristic quantities. The most ener-getic wave number κ0 = 4 in the incomingturbulence field is taken as the characteris-tic length scale.The mean conservation equations

along with the k − � and u�e� modelequations are solved using the finitevolume approach, where the equa-tions are written in a vector form.

∂

∂t

ρρ �u�Eρ kρ �

ρu�e�

� �� �[U]

+∂

∂x

ρ �uρ �u2 + p

�u( �E + p)ρ �u kρ �u �

ρ �uu�e�

� �� �[F]

=

000

−(2/3)ρk(∂�u/∂x)(1− b�1)− ρ�−(2/3)c1ρ�(∂�u/∂x)− c�2ρ(�

2/k)

cue(k/�u)cp(∂ �T/∂x)− c2(u�e�/L�)

� �� �[S]

.(22)

Here, [U] is the vector of the conserved vari-ables, [F] is the vector containing the in-viscid fluxes and [S] is the vector contain-ing the source terms, where tilde representsFavre averaging and �E is the mean total en-ergy.

The inviscid fluxes at cell faces are eval-uated using the Lax-Friedrich scheme andthe turbulent source terms are evaluated atthe cell centers. The mean flow gradients

in the production terms are discretized us-ing a second-order accurate central differ-ence scheme. Explicit time integration isachieved using the first order forward Eulermethod. The Lax-Friedrich method is moredissipative than the modified Steger Warm-ing method used by Sinha and Balasridhar[25], but can produce identical results withlarger number of grid points.

We simulate shock-turbulence interactionfor varying shock-strength, i.e. Mach num-

13

ber ranging from low supersonic to hyper-sonic values (see table 1). The turbu-lent Mach number Mt just upstream of theshock wave is taken as 0.22. The normal-ized turbulence kinetic energy is thus givenby k1 = M2

t /2, where subscript 1 repre-sents shock-upstream values. The Reynoldsnumber based on Taylor microscale Reλ =λu

�rms/ν = 40 [26], where u

�rms =

�2k/3.

The Taylor microscale is given by λ =�10νk/�, where ν is the mean kinematic

viscosity of the fluid. The turbulent dissi-pation rate � can thus be computed as

�1 =5√3

�M3

t

κ0λ

�1

Reλ, (23)

where κ0λ = 0.84.The computational domain extends from

κ◦x = −7.6 to 29.6, with the shock centeredat x = 0. The homogeneous turbulence up-stream of the shock follows decay relations

�udkdx

= −�,

�u d�dx

= −c�2�2

k,

with solution given by

k0 = k1

�1 + κ0xsh

(c�2 − 1)

M1

�0k0

� 1c�2−1

,

�0 = �1

�1 + κ0xsh

(c�2 − 1)

M1

�0k0

� c�2c�2−1

,

(24)

where k0 and �0 are the inlet values obtainedfrom the shock upstream values (see table1). The same values are used to initializethe turbulence variables in the simulation.Here, c�2 = 1.2 and κ0xsh = 6.83. The meanflow quantities are initialized to Rankine-Hugoniot jump conditions at the shock atx = 0. At the exit boundary, we specifyzero gradient boundary condition for all themean flow and turbulence variables.

5.2. Numerical results

The mean flow and the model equationsare solved using 800 points equally spacedalong the shock-normal direction. Thejump in the turbulent energy flux for vary-ing Mach numbers is presented in figure 9.For low Mach numbers upto M1 = 2, thereis a good match between the Lax-Friedrichand the RK4 solution reproduced from fig-ure 7. However, at higher Mach numbers,the numerical scheme overpredicts the jumpin the turbulent energy flux. The reason forthis can be attributed to the error in thenumerical solution computed at the shock.The earlier results obtained for a specifiedtangent hyperbolic shock profile is devoidof such numerical error. The numerical er-ror is not found to decrease with successivegrid refinement; see results obtained using400, 800 and 1200 grid points in figure 9. Asimilar trend is observed with the k − � so-lution at shock waves [25] and is discussedbelow. The streamwise variation of the tur-bulent energy flux obtained by solving (21)numerically is shown in figure 8. The re-sults are compared with the RK4 methodand the corresponding DNS data. For thelow Mach number case of M1 = 1.50, boththe RK4 and the numerical solution matchwell and are comparable to the DNS data.With increasing Mach number, the numer-ical solution deviates from the RK4 resultboth in the jump across the shock and thedownstream evolution. This deviation onceagain can be attributed to the nonconserva-tive error in the source terms of the trans-port equation for k, � and u�e�.Sinha and Balasridhar [25] studied

the numerical issues involved while solv-ing shock-turbulence interaction using theshock-unsteadiness k-� equations. It wasfound that for shock capturing simulations,like those in this work, the nonconservative

14

*

*

*

*

*

*

M1

1 3 5 7-0.5

0

0.5

1

1.5

2

2.5

LIA

NC-1200

RK4

NC-800

NC-400

Figure 9: The post-shock turbulent energy flux ascomputed using the Lax-Friedrich scheme appliedto the nonconservative (NC) form of the modelequations, compared with the Runge-Kutta (RK4)method and linear theory (LIA) predictions. Solu-tion computed on different grid sizes: 400 (gradi-ent), 800 (triangle), 1200 (star). Normalisation asdescribed in figure 2.

nature of the production terms can result inlarge numerical error at shock waves. Theinviscid flux terms, on the other hand, arein conservative form. In a finite-volume ap-proach, the fluxes are evaluated at cell in-terfaces. The corresponding truncation er-ror involves higher-order derivatives of theflow variables, which take large values in theshock wave. It is shown that the error fromadjacent cells cancel at the common inter-faces when all the finite volume cells span-ning the shock wave are taken together. Thenet error across the shock wave is thereforea function of the higher-order derivativesevaluated in a region of smooth flow solu-tion, outside the shock wave. On successivegrid refinement, the leading-error term van-ishes and the solution tends towards the ex-act integral. Further details can be found inthe appendix of Sinha and Balasridhar [25].

The source terms in both the k and �equations involve flow derivatives that areevaluated using a symmetric second-orderdiscretization Sinha and Balasridhar [25].The corresponding truncation error involvesthird and higher-order derivatives of veloc-ity, pressure and temperature. On integra-tion over a finite volume cell, the corre-sponding error term thus involves integra-tion of these higher order derivatives. Theleading error term scales as 1/δ3, where δis the computed shock thickness, and hencethe error takes large values in the region ofa flow discontinuity. When all the finite vol-ume cells spanning the shock wave are takentogether, these error terms from the indi-vidual cells add up and appear as numeri-cal volume sources. Unlike the conservativeflux terms, the telescopic effect of error can-cellation between adjacent cells is absent inthe nonconservative source terms [25]. Thisleads to significant error that are compara-ble or larger than the physical production atthe shock waves. Also, the numerical errordoes not decrease in magnitude with succes-sive grid refinement as observed for smoothsolutions.

To restrict the truncation errors at shockwaves, Sinha and Balasridhar [25] pro-poses an alternative form of the shock-unsteadiness k-� equations obtained bytransforming the turbulence variables. Con-served quantities involving k and � are for-mulated and transport equations for thesenew variables are derived. The new k − �equations capture identical physics as theoriginal shock-unsteadiness model, but arefree of the nonconservative source terms,and therefore show dramatic improvementin numerical accuracy. Here, we follow asimilar approach for the u�e� model equa-tion given in (21) to derive the correspond-ing conservative form.

15

5.3. Conservative formulationA conservative formulation of the govern-

ing equations for k and � is derived by Sinhaand Balasridhar [25], and is given as

ρ �u∂f∂x

= −ρ�ρ−(2/3)co , (25)

ρ �u∂g∂x

=−c�2�

2

kρ1−(2/3)c1 , (26)

where co = 1− b�1 and b�1 = 0.4(1− exp(1−M1)), and the transformed variables f =

kρ−23co and g = �ρ−

23c1 are based on the

closed form solution of the original equa-tions. Solving the above two equations in-stead of their corresponding nonconserva-tive forms significantly improved the pre-diction of TKE and its dissipation rate incanonical STI; see Sinha and Balasridhar[25] for details. A closed form solution for

M1

1 3 5 7-0.5

0

0.5

1

1.5

2

2.5

LIA

Conservative

closed-form

solution

Nonconservative

Figure 10: Comparison of the turbulent energy fluxcomputed using the Lax-Friedrich scheme appliedto the conservative and nonconservative forms ofthe model equations with the closed-form solutionand the far-field LIA values. Normalisation as de-scribed in figure 2.

energy flux can be obtained by writing (19)as

∂

∂x

�u�e�

�=

cuecpρ�u fρ(2c0/3+1)∂

�T∂x

, (27)

where f is a conserved variable across theshock [25]. This equation reduces to

∂

∂x

�u�e�

�= −cuef(ρ �u)2c0/3�u−2c0/3

∂�u∂x

,

(28)using the total energy conservation acrossa shock wave. Integrating between theshock-upstream (subscript 1) and shock-downstream (subscript 2) locations yields

u�2e

�2 = A1

��u1−2c0/32 − �u1−2c0/3

1

�, (29)

where A1 = −cuef(ρ�u)2c0/3/(1− 2c0/3) is aconserved quantity across the shock. Thevalue of the energy flux predicted by thisclosed form solution is plotted for varyingupstream Mach numbers in figure 10. Agood match is observed with the solutionobtained by solving the corresponding dif-ferential equation (19) using RK4 methodas shown in figure 9.The closed form solution is used to define

a new variable

h = u�e� − A1�u1−2c0/3, (30)

and write the model equation for u�e� as

∂h

∂x= −c2

u�e�

L�

, (31)

which captures the same physics as theoriginal equation (21), but is numericallymore robust at shock waves than the for-mer. This is because the nonconservativesource terms involving the mean tempera-ture derivative is eliminated from the trans-port equation. Numerical implementationof the conservative model equations followthe same method as described in (22), withk, � and u�e� replaced by f , g and h, respec-tively. The source term vector [S] is ap-propriately modified as per the right-hand

16

sides of (25), (26) and (31). The initial andboundary conditions are recast in terms ofthe transformed variables, and the turbu-lence variables k, � and u�e� are obtainedfrom the solution by invoking the reversetransformation.Figure 10 shows the peak turbulent en-

ergy flux for varying upstream Mach num-ber obtained by numerically solving theconservative form of the model equations.The results match exactly with the closedform solution and are a significant improve-ment over the nonconservative results. Fig-ure 11 shows the streamwise evolution of theturbulent energy flux computed on differentnumerical grids. The data corresponds tothe higher Mach number cases, where thenonconservative error is most prominent.The results show a systematic grid conver-gence, with the difference in the peak u�e�

values diminishing with successive grid re-finement. The model predictions also com-pare well with the DNS data up to x � L�

for the Mach 2.5 interaction and upto x �0.7L� for the high Mach number case. Tur-bulent energy flux predicted for the Mach1.5 and 1.87 flows are almost identical tothose in figure 8, and they are also compa-rable to the post-shock DNS data.Figure 10 shows the peak turbulent en-

ergy flux for varying upstream Mach num-ber obtained by numerically solving theconservative form of the model equations.The results match exactly with the closedform solution and are a significant improve-ment over the nonconservative results. Fig-ure 11 shows the streamwise evolution of theturbulent energy flux computed on differentnumerical grids. The data corresponds tothe higher Mach number cases, where thenonconservative error is most prominent.The results show a systematic grid conver-gence, with the difference in the peak u�e�

values diminishing with successive grid re-finement. The model predictions also com-pare well with the DNS data up to x � L�

for the Mach 2.5 interaction and upto x �0.7L� for the high Mach number case. Tur-bulent energy flux predicted for the Mach1.5 and 1.87 flows are almost identical tothose in figure 8, and they are also compa-rable to the post-shock DNS data.

6. Conclusion

In this paper, we look at various mod-elling strategies to predict the turbulentenergy flux generated in canonical shock-turbulence interaction. Application of con-ventional k − � models based on the gra-dient diffusion hypothesis yields large neg-ative value of energy flux in the shock re-gion, which rises drastically with increasingMach number and grid refinement. Thesevalues are much higher in magnitude thanthe post-shock DNS predictions, which yielda low positive energy flux. Realizable modelalso predicts negative values of the energyflux in the shock region, but the peak valueis limited by the realizability constraint. Anew model is proposed using results fromlinear interaction analysis, which has beenshown to capture the key physics of tur-bulent energy flux generated at a shockwave. The model is based on a heat-fluxlimiter formulation similar in form to therealizability constraint and is able to pre-dict the peak turbulent energy flux overa range of Mach numbers. We also de-velop a transport equation model for theturbulent energy flux, based on the linearand inviscid interaction of turbulent fluctu-ations with a shock wave. The model pa-rameters are taken from the well-establishedshock-unsteadiness k − � model,and a nu-merically robust form is proposed by elimi-

17

x/Lε

0 0.5 1 1.5 2

0

0.15

0.3

M1=1.50

RK4

Consv.

Nonconsv.

(a)

x/Lε

0 0.5 1 1.5 2

0

0.2

0.4

M1=1.87

(b)

x/Lε

-0.5 0 0.5 1 1.5 2

0

0.2

0.4

0.6M

1= 2.5

(c)

x/Lε

-0.5 0 0.5 1 1.5 2

0

0.25

0.5

0.75

1

M1= 3.5

(d)

Figure 11: Streamwise variation of u�e� as computed using the conservative form of the model equations arecompared with the DNS data (symbols) for two upstream Mach numbers. The vertical lines near x/L� = 0represent the mean shock thickness, and the Lax-Friedrich scheme is employed on three different grids: 400(dash dot dot), 800 (dash) and 1200 (solid). Normalisation as described in figure 2.

nating the nonconservative source terms inthe equation. The new physics-based modelis found to predict DNS values of the peakenergy flux correlation and the acoustic de-cay downstream of the shock wave.

References

[1] D. V. Gaitonde, Progress in shockwave/boundary layer interactions, Progress inAerospace Sciences 72 (2015) 80 – 99.

[2] V. Pasquariello, M. Grilli, S. Hickel, N. A.Adams, Large-eddy simulation of passive

shock-wave/boundary-layer interaction con-trol, International Journal of Heat and FluidFlow 49 (2014) 116 – 127.

[3] J. Quaatz, M. Giglmaier, S. Hickel, N. Adams,Large-eddy simulation of a pseudo-shock sys-tem in a Laval nozzle, International Journal ofHeat and Fluid Flow 49 (2014) 108 – 115.

[4] J. L. Brown, Turbulence Model Validationfor Hypersonic Flows, 8th AIAA/ASME JointThermophysics and Heat Transfer Conference,St. Louis, MO .

[5] C. J. Roy, F. G. Blottner, Review and as-sessment of turbulence models for hypersonicflows, Progress in Aerospace Sciences 42 (7)

18

(2001) 469–530.[6] X. Xiao, H. A. Hassan, J. R. Edwards, R. L. G.

Jr., Role of turbulent Prandtl numbers on heatflux at hypersonic Mach numbers, AIAA Jour-nal 45 (4) (2007) 806–813.

[7] T. P. Sommer, R. M. C. So, H. S. Zhang,Near-Wall Variable-Prandtl-Number Turbu-lence Model for Compressible Flows, AIAAJournal 31 (1) (1993) 27–35.

[8] U. C. Goldberg, S. Palaniswamy, P. Batten,V. Gupta, Variable turbulent Schmidt andPrandtl number modeling, Engineering ap-plication of computational fluid mechanics 4(2010) 511.

[9] M. Aouissi, A. Bounif, K. Bensayah, Scalarturbulence model investigation with variableturbulent Prandtl number in heated jets anddiffusion flames, Heat Mass Transfer 44 (2008)1065.

[10] K. Abe, K. Suga, Towards the developmentof a Reynolds-averaged algebraic turbulentscalar-flux model, International Journal ofHeat and Fluid Flow 22 (2001) 19–29.

[11] R. D. W. Bowersox, Extension of equilibriumturbulent heat flux models to high-speed shearflows, J. Fluid Mech. 633 (2009) 61–70.

[12] R. Quadros, K. Sinha, J. Larsson, Turbulentenergy flux generated by shock/homogeneous-turbulence interaction, accepted in J. FluidMech. .

[13] L. S. G. Kovasznay, Turbulence in supersonicflow, J. Aeronaut. Sci. 20 (10) (1953) 657–674.

[14] J. Larsson, I. Bermejo-Moreno, S. K. Lele,Reynolds- and Mach number effects in canon-ical shock-turbulence interaction, J. FluidMech. 717 (2013) 293–321.

[15] K. Sinha, K. Mahesh, G. V. Candler, Model-ing shock-unsteadiness in shock-turbulence in-teraction, Phys. Fluids 15 (2003) 2290–2297.

[16] K. Sinha, Evolution of enstrophy inshock/homogeneous turbulence interaction,J. Fluid Mech. 707 (2012) 74–110.

[17] K. Sinha, K. Mahesh, G. V. Candler, Model-

ing the effect of shock unsteadiness in shock-wave / turbulent boundary-layer interactions,AIAA Journal 43 (3) (2005) 586–594.

[18] A. A. Pasha, K. Sinha, Shock-unsteadinessmodel applied to oblique shock-wave/turbulent boundary-layer interaction,Int. J. Comput. Fluid Dyn. 22 (8) (2008)569–582.

[19] A. A. Pasha, K. Sinha, Shock-unsteadinessmodel applied to hypersonic shock-wave/turbulent boundary-layer interactions,Journal of Propulsion and Power 28 (1) (2012)46–60.

[20] R. Quadros, K. Sinha, J. Larsson, Kovasznaymode decomposition of velocity-temperaturecorrelation in canonical shock-turbulence in-teraction, accepted in Flow, Turbulence andCombustion .

[21] K. Mahesh, S. K. Lele, P. Moin, The influenceof entropy fluctuations on the interaction ofturbulence with a shock wave, J. Fluid Mech.334 (1997) 353–379.

[22] K. Mahesh, S. K. Lele, P. Moin, The interac-tion of a shock wave with a turbulent shearflow, Tech. Rep. 69, Thermosciences division,Department of Mechanical Engineering, Stan-ford University, Stanford, CA., 1996.

[23] F. Thivet, D. D. Knight, A. A. Zhel-tovodov, A. I. Maksimov, Importance of limit-ing the turbulent stresses to predict 3D shock-wave/boundary-layer interactions, 23rd Inter-national Symposium on Shock Waves, FortWorth, TX .

[24] V. K. Veera, K. Sinha, Modelling the ef-fect of upstream temperature fluctuationson shock/homogeneous turbulence interaction,Phys. Fluids 21 (2009) 025101.

[25] K. Sinha, S. J. Balasridhar, Conservative for-mulation of the k − � turbulence model forshock-turbulence interaction, AIAA Journal51 (8) (2013) 1872–1882.

[26] H. Tennekes, J. L. Lumley, in: A First Coursein Turbulence, The MIT press, 1972.

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