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Supersonic turbulent ﬂows over sinusoidal

rough walls

Mostafa Aghaei-Jouybari1, Junlin Yuan2†, Zhaorui Li3,

Giles J. Brereton2, and Farhad A. Jaberi2

1Department of Mechanical Engineering, Johns Hopkins University, Baltimore,

MD 21218, USA

2Department of Mechanical Engineering, Michigan State University, East Lansing,

MI 48824, USA

3Department of Engineering, Texas A&M University-Corpus Christi, Corpus Christi,

TX 78412, USA

(Received xx; revised xx; accepted xx)

Direct numerical simulations were performed to characterize fully developed supersonic

turbulent channel ﬂows over isothermal rough walls. The eﬀect of roughness was incorpo-

rated using a level-set/volume-of-ﬂuid immersed boundary method. Turbulence statistics

of ﬁve channel ﬂows are compared, including one reference case with both walls smooth

and four cases with smooth top walls and bottom walls with two-dimensional (2D) and

three-dimensional (3D) sinusoidal roughnesses. Results reveal a strong dependence of the

turbulence on the roughness topography and the associated shock patterns. Speciﬁcally,

the 2D geometries generate strong oblique shock waves that propagate across the channel

and are reﬂected back to the rough-wall side. These strong shocks are absent in the

smooth-wall channel and are signiﬁcantly weaker in cases with 3D roughness geometries,

replaced by weak shocklets. At the impingement locations of the shocks on the top wall

in the 2D roughness cases, localized augmentations of turbulence shear production are

observed. Such regions of augmented production also exist for the 3D cases, at a much

weaker level. The oblique shock waves are thought to be responsible for a more signiﬁcant

entropy generation for cases with 2D surfaces than those with 3D ones, leading to a higher

irreversible heat generation and consequently higher temperature values in 2D roughness

cases. In the present supersonic channels, the eﬀects of roughness extend beyond the

near-wall layer due to the shocks. This suggests that outer layer similarity may not fully

apply to a rough-wall supersonic turbulent ﬂow.

1. Introduction

The eﬀects of wall roughness on physics, control, and modeling of compressible ﬂows

(subsonic, sonic, super- and hypersonic) are not well understood today. An understanding

of these eﬀects is important for ﬂight control and thermal management of high speed

vehicles, especially for reentry applications and reusable launch vehicles. In high speed

ﬂow studies, roughness is typically modeled with an isolated (e.g. steps, joints, gaps,

etc.), or a distributed (e.g. screw threads, surface ﬁnishing, and ablation) organized

structure. Reda (2002) and Schneider (2008) have reviewed the eﬀects of roughness on

boundary layer transition, based on experimental wind-tunnel and in-ﬂight test data

of ﬂows in supersonic and hypersonic conditions. Radeztsky et al. (1999) analyzed the

eﬀects of roughness of a characteristic size of 1-µm (i.e. a typical surface ﬁnish) on

†Email address for correspondence: junlin@egr.msu.edu

2M. Aghaei-Jouybari et al.

transitions in swept-wing ﬂows, and Latin (1998) investigated eﬀects of roughness on

supersonic boundary layers using rough surfaces with an equivalent sandgrain height ks

of O(1mm), corresponding to 100 < k+

s<600 (where superscript + shows normalization

in wall units). Experimental studies of distributed roughness eﬀects on compressible ﬂows,

boundary layer transition, and heat transfer include those of Braslow & Knox (1958);

Reshotko & Tumin (2004); Ji et al. (2006) and Reda et al. (2008), among others.

There are ample studies in the literature focusing on the dynamics, modeling, statistics

and structures of ﬂows over rough walls in incompressible regime (Nikuradse 1933;

Raupach et al. 1991; Jim´enez 2004; Mejia-Alvarez & Christensen 2010; Volino et al. 2011;

Talapatra & Katz 2012; Yang et al. 2015; Flack 2018; Thakkar et al. 2018; Yuan & Aghaei-

Jouybari 2018; Ma et al. 2021; Aghaei-Jouybari et al. 2021, 2022). Here, however, we

focus on the compressible (mostly supersonic) ﬂows over rough walls, the understanding

of which is much limited. Some of these studies are summarized below.

Tyson & Sandham (2013) analyzed supersonic channel ﬂows over 2D sinusoidal rough-

ness at Mach number (M) of M= 0.3, 1.5 and 3 to understand compressibility eﬀects

on mean and turbulence properties across the channel. They used body-ﬁtted grids

to perform the simulations and found that the values of velocity deﬁcit decrease with

increasing Mach number. Their results suggest strong alternation of mean and turbulence

statistics due to the shock patterns associated with the roughness.

Ekoto et al. (2008) experimentally investigated the eﬀects of square and diamond

roughness elements on the supersonic turbulent boundary layers, to understand how

roughness topography alters the local strain-rate distortion, dmax, which has a direct

eﬀect on turbulence production. Their results indicated that the surface with d-type

square roughness generate weak bow shocks upstream of the cube elements, causing small

dmax (≈ −0.01), and the surface with diamond elements generate strong oblique shocks

and expansion waves near the elements, causing a large variation in dmax (ranging from

−0.3 to 0.4 across the elements). Studies of Latin (1998), Latin & Bowersox (2000) and

Latin & Bowersox (2002) included a comprehensive investigation on supersonic turbulent

boundary layers over rough walls. Five rough surfaces (including a 2D bar, a 3D cube,

and three diﬀerent sandgrain roughness) have been analyzed at M= 2.9. Eﬀects of wall

roughness on mean ﬂow, turbulence, energy spectra and ﬂow structures were studied.

Their results showed strong linear dependence of turbulence statistics on the surface

roughness, and also, strong dependencies of turbulent length scales and inclination angle

of coherent motions on the roughness topography. Muppidi & Mahesh (2012) analyzed

the role of ideal distributed roughness on the transition to turbulence in supersonic

boundary layers. They found that counter-rotating vortices, generated by the roughness

elements, break the overhead shear layer, leading to an earlier transition to turbulence

than on a smooth wall. A similar study was conducted by Bernardini et al. (2012), who

investigated the role of isolated cubical roughness on boundary layer transition. Their

results suggest that the interaction between hairpin structures, shed by the roughness

element, and the shear layer expedites transition to turbulence, regardless of the Mach

number. Recently, Modesti et al. (2022) analyzed compressible channel ﬂows over cubical

rough wall, both in transitionally and fully rough regimes, and found that the logarithmic

velocity deﬁcit (with respect to the baseline smooth wall) depends strongly on the

Mach number, and one must employ compressibility transformations to account for the

compressibility eﬀects and to establish an appropriate analogy between compressible and

incompressible regimes.

Despite ﬁndings in these studies, comprehensive understanding of turbulence statics

and dynamics, as well as their connection to roughness geometry (especially for dis-

tributed roughnesses) and the associated compressibility eﬀects remained unavailable.

Supersonic ﬂows over rough walls 3

Most numerical studies on this topic have focused on isolated roughness Bernardini et al.

(see e.g. 2012) or ideal distributed roughness such as wavy walls Tyson & Sandham

(see e.g. 2013), due to the simplicity in mesh generation and numerical procedures.

However, complex distributed roughness is of primary importance and more relevant

to ﬂight vehicles, since in high-speed ﬂows “even the most well-controlled surface will

appear rough as the viscous scale becomes suﬃciently small” (Marusic et al. 2010). Also,

according to Schneider (2008), real vehicles may develop surface roughness during the

ﬂight which is not present before launch. This ﬂight-induced roughness may be discrete

steps and gaps on surfaces from thermal expansion, or distributed roughness induced by

ablation or the impact of dust, water, or ice droplets.

These studies demonstrate the need for better understanding the eﬀects of complex

distributed rough surfaces. Numerical studies of ﬂow over complex roughness geometries

beneﬁt from the immersed boundary (IB) method (see a detailed review by Mittal

& Iaccarino 2005), which has multiple advantages compared to the employment of a

body conformal grid, primarily in the ease of mesh generation. A summary of numerical

methods based on immersed boundaries in the compressible ﬂow literature is given below.

Ghias et al. (2007) used ghost cell method to simulate 2D viscous subsonic compressible

ﬂows. They imposed the Dirichlet boundary condition (BC) for velocity (u) and temper-

ature (T) on the immersed boundaries. The pressure (P) at the boundary was obtained

using the equation of state and the value of density (ρ) was obtained through extrap-

olation. Their method was second-order accurate, both locally and globally. Chaudhuri

et al. (2011) used the ghost cell method to simulate 2D inviscid, sub- and supersonic

compressible ﬂows. They applied direct forcing for ρ,uand total energy (E) equations,

while Pwas determined based on the equation of state. They used a ﬁfth-order-accurate

WENO shock-capturing scheme by using two layers of ghost cells. Yuan & Zhong (2018)

also used ghost cell method to simulate 2D (sub- and supersonic) compressible ﬂows

around moving bodies. Vitturi et al. (2007) used a discretized forcing approach for a

ﬁnite volume solver to simulate 2D/3D viscous subsonic multiphase compressible ﬂows;

the forcing term was determined based on an interpolation procedure. They imposed

Dirichlet BC for uand T; the equation of state was used for Pand ﬂux correction for ρ

and E. Wang et al. (2017) used continuous forcing (or penalty IB method) to simulate

ﬂuid-structure interaction with 2D compressible (sub, super, and hyper sonic) multiphase

ﬂows.

Most of these studies used sharp-interface IB methods, which allows the boundary

conditions at the interface to be imposed exactly. However, 3D ﬂows with complex

interface geometries (especially with moving interfaces) cause diﬃculties and require

special considerations. Speciﬁcally, issues arise when there are multiple image points for

a ghost cell, or when there is none. Luo et al. (2017) addressed some of these issues in

2D domains. In addition, the interpolation schemes are dependent on the ghost point

locations in the solid domain; the situation becomes complicated for 3D domains. To

account for these diﬃculties in 3D ﬂows with complex interface geometries, the boundary

values can be imposed through a prescribed distribution across the interface, instead

of being imposed precisely. Examples include the approaches based on ﬂuid-volume-

fraction weighting proposed by Fadlun et al. (2000) and Scotti (2006), developed for

incompressible ﬂows.

In this study we ﬁrst introduce a compressible-ﬂow IB method that is a combination

of both volume-of-ﬂuid (VOF) and level-set methods—a modiﬁed version of the level-set

methods used in multi-phase ﬂow simulations in the incompressible regimes (Sussman

et al. 1994, 1999) and compressible regimes (Li et al. 2008), as well as the VOF method

of Scotti (2006) for incompressible rough wall ﬂows—and validate it by comparing mean

4M. Aghaei-Jouybari et al.

and turbulence statistics with a baseline simulation using a body-ﬁtted mesh. Then

we analyze the simulation results in supersonic channel ﬂows at M= 1.5 and a bulk

Reynolds number of 3000 (based on the channel half height) over two 2D and two 3D

sinusoidal surfaces. Analyses are ﬁrst carried out with respect to the mean quantities and

turbulent statistics for an overall comparison of the ﬂow ﬁelds, then with the transport

equations of Reynolds stresses to compare turbulence production and transports. Finally,

we perform conditional analyses for energy budget attributed to solenoidal, compression

and expansion regions of the ﬂow. Section 2 describes the governing equations, numerical

setup and the IBM formulation. Results of the mean ﬂow and turbulence statistics,

Reynolds stress budgets and conditional analysis are explained in section 3, and the

manuscript is concluded in section 4.

2. Numerical setup and formulation

2.1. Governing equations

The non-dimensional forms of compressible Navier-Stokes equations are

∂ρ

∂t +∂

∂xi

(ρui)=0,(2.1a)

∂ρui

∂t +∂

∂xjρuiuj+pδij −1

Reτij =f1δi1,(2.1b)

∂E

∂t +∂

∂xiui(E+p)−1

Reujτij +1

(γ−1)PrReM2qi=f1u1,(2.1c)

where x1,x2,x3(or x,y,z) are coordinates in the streamwise, wall-normal and spanwise

directions, with corresponding velocities of u1,u2and u3(or u,vand w). Density,

pressure, temperature and dynamic viscosity are denoted by ρ,p,Tand µ, respectively.

E=p/(γ−1) + ρuiui/2 is the total energy, γ≡Cp/Cvis the ratio of speciﬁc heats

(assumed to be 1.4), τij =µ∂ui

∂xj+∂uj

∂xi−2

3

∂uk

∂xkδij is the viscous stress tensor, and

qi=−µ∂T

∂xiis the thermal heat ﬂux. f1is a body force that drives the ﬂow in the

streamwise direction, analogous to the pressure gradient. All these variables are in

their non-dimensional forms. The reference Reynolds, Mach and Prandtl numbers are,

respectively, Re ≡ρrUrLr/µr,M≡Ur/√γRTr, and Pr ≡µrCp/κc, where subscript r

stands for reference values (to be deﬁned in section 2.3). The gas constant Rand the

speciﬁc heats Cpand Cvare assumed to be constant throughout the domain (calorically

perfect gas). They are related by R=Cp−Cv. The heat conductivity coeﬃcient is

denoted by κc.

The set of equations in (2.1) is closed through the equation of state, which for a perfect

gas is

p=ρT

γM 2.(2.2)

Equations (2.1) and (2.2) are solved using a ﬁnite-diﬀerence method in a conservative

format and a generalized coordinate system. A ﬁfth-order monotonicity-preserving (MP)

shock-capturing scheme and a sixth order compact scheme are utilized for calculating

the inviscid and viscous ﬂuxes, respectively. The solver uses local Lax-Friedriches (LLF)

ﬂux-splitting method and employs an explicit third-order Runge-Kutta scheme for time

advancement. The computational solver has been used for direct numerical simulation

(DNS) and large-eddy simulation (LES) of a range of compressible turbulent ﬂows

Supersonic ﬂows over rough walls 5

including those involving smooth surfaces (Tian et al. 2017, 2019; Jammalamadaka et al.

2013, 2014; Jammalamadaka & Jaberi 2015). Readers are referred to Li & Jaberi (2012)

for extensive details of the compressible solver.

2.2. Details of the present IB method

The present IB method is a combination of level-set (Sussman et al. 1994; Gibou et al.

2018; Li et al. 2008) and volume-of-ﬂuid (VOF, Scotti 2006) methods. It is designed for

stationary interfaces only. The level-set ﬁeld ψ(x, y, z) is deﬁned as the signed distance

function to the ﬂuid-solid interface. Based on the prescribed roughness geometry, the ψ

ﬁeld is obtained by diﬀusing an initial discontinuous marker function,

ψ0(x, y, z) =

1 in ﬂuid cells,

0 in interface cells,

−1 in solid cells,

(2.3)

in the interface-normal direction until a narrow band along the interface, within which

ψis sign-distanced, is generated; this is similar to the reinitialization process conducted

by Sussman et al. (1994), and done by solving

∂ψ

∂τ = sign(ψ)(1 − |∇ψ|),(2.4)

where τis a ﬁctitious time controlling the width of the interface band. It is suﬃcient to

march in (ﬁctitious) time until a band width of up to 2-3 grid size is obtained.

Based on the level-set ﬁeld, the VOF ﬁeld, ϕ(x, y, z), is constructed as

ϕ≡(1 + ψ)/2,(2.5)

such that ϕ= 0, 0 <ϕ<1, and ϕ= 1 correspond to the solid, interface and ﬂuid cells,

respectively.

To impose the desired boundary condition for a test variable θ(x, y, z, t), we correct

the values of the variable at the beginning of each Runge-Kutta substep. The correction

is similar to the approach used by Scotti (2006) and that of Yuan and co-workers (Yuan

& Piomelli 2014, 2015; Yuan et al. 2019; Shen et al. 2020; Mangavelli et al. 2021), i.e.,

θ→ϕθ + (1 −ϕ)θb,(2.6)

for Dirichlet BC and

∂θ

∂n =∇θ·b

n=∂θ

∂n b

(2.7)

for Neumann BC, where the subscript bdenotes boundary values and b

nis the unit normal

vector pointing into the ﬂuid region at the interface. b

nis obtained as

b

n=∇ψ=∇ϕ/|∇ϕ|.(2.8)

Note that ϕ(x, y, z) does not represent exactly the ﬂuid volume fraction in each grid cell.

Instead, ϕis termed the VOF ﬁeld because of the analogy between the BC imposition in

equations (2.6) and (2.7) and the approach of Scotti (2006) using the exact volume-of-

ﬂuid. As will be shown in section 2.4, the accuracy of the IB method herein is suﬃcient

to produce matching single-point statistics compared to a simulation using body-ﬁtted

grid.

6M. Aghaei-Jouybari et al.

Figure 1. Surface roughnesses.

2.3. Surface roughnesses and simulation parameters

Fully developed, periodic compressible channel ﬂows are simulated using four roughness

topographies. The channels are roughened only at one surface (bottom wall) and the other

surface is smooth. A reference smooth-wall channel is also simulated for validation and

comparison purposes. The channel dimensions in streamwise, wall-normal and spanwise

directions are, respectively, Lx= 12δ,Ly= 2δand Lz= 6δ, where δis the channel

half-height.

Figure 1 shows four roughness topographies used for the present simulations. These

rough-wall cases are C1-C4, and the smooth-wall baseline case is denoted as SM. All

rough cases share the same crest height, kc= 0.1δ. The trough location is set at y= 0.

Cases C1 and C2 are 2D sinusoidal surfaces with streamwise wave-lengths of λx= 2δ

and λx=δ, respectively. The roughness heights, k(x, z), for these surfaces are prescribed

as

k(x, z)/δ = 0.051 + cos(2πx/λx).(2.9)

Cases C3 and C4 are 3D sinusoidal surfaces with equal streamwise and spanwise wave-

lengths of (λx, λz) = (2δ, 2δ) for C3, and (λx, λz)=(δ, δ) for C4. The roughness heights

for them are prescribed as

k(x, z)/δ = 0.051 + cos(2πx/λx) cos(2πz/λz).(2.10)

Table 1 summarizes some statistical properties of the surface geometries. These statistics

are various moments of surface height and surface eﬀective slopes.

For a test variable θ, the time, Favre and spatial averaging operators are denoted

respectively by θ,e

θ=ρθ/ρ and θ. Intrinsic planar averaging is used for the spatial

averaging, where a y-dependent ﬂuid variable is averaged per unit ﬂuid planar area,

θ= 1/AfRAfθdA;Af(y) is the area of ﬂuid at a given y. At y= 0, Af= 0 since all

area is inside solid. As a result, data at y= 0 are not included in intrinsically averaged

wall-normal proﬁles. Fluctuation components θ′,θ′′ and θ′′′ are deﬁned following the

Supersonic ﬂows over rough walls 7

Case kckavg krms RaExEzSkKu

C1 0.1 0.05 0.035 0.032 0.100 0.000 0.0 1.50

C2 0.1 0.05 0.035 0.032 0.200 0.000 0.0 1.50

C3 0.1 0.05 0.025 0.020 0.064 0.064 0.0 2.25

C4 0.1 0.05 0.025 0.020 0.127 0.127 0.0 2.25

Table 1. Statistical parameters of roughness topography. kcis the peak-to-trough height,

kavg =1

AtRx,z k(x, z)dA is the average height, krms =q1

AtRx,z(k−kav g )2dA is the

root-mean-square (r.m.s.) of roughness height ﬂuctuation, Ra=1

AtRx,z |k−kavg |dA is

the ﬁrst-order moment of height ﬂuctuations, Exi=1

AtRx,z

∂k

∂xi

dA is the eﬀective

slope in the xidirection, Sk=1

AtRx,z(k−kav g )3dA.k3

rms is the height skewness, and

Ku=1

AtRx,z(k−kav g )4dA.k4

rms is the height kurtosis. Here, k(x, z) is the roughness height

distribution; Atthe total planar areas of channel wall. Values of kc,kavg,krms and Raare

normalized by δ.

triple decomposition of Raupach & Shaw (1982), such that

θ=θ+θ′

=e

θ+θ′′

=θ+θ′′′.

(2.11)

Periodic BCs are used in the streamwise and spanwise directions. A no-slip iso-thermal

wall BC is imposed at both top and bottom walls. The values of velocity and temperature

on both walls (denoted by subscript w) are uw=0and Tw= 1 (the temperature at the

wall is used as the reference temperature, i.e. Tr=Tw). There is no need to impose a BC

for density; equation (2.1a) is solved using third-order accurate one-sided diﬀerentiation

to update the density values at the boundaries. This approach is similar to those used

in other wall-bounded compressible ﬂow studies (see e.g. Coleman et al. 1995; Tyson &

Sandham 2013). The pressure at the boundaries is calculated using the equation of state.

The reference density and velocity used here are those of bulk values, deﬁned as ρr≡

1

VfRVfρdvand Ur≡1

ρrVfRVfρudv(where Vfis the ﬂuid occupied volume). The reference

length scale is δ. All these values are set to be 1 here. The time-dependent body force

f1in NS equation (2.1) is adjusted automatically in each time-step to yield the constant

bulk velocity under the prescribed Reynolds number. Speciﬁcally, at each time-step the

bulk velocity Uris ﬁrst calculated and f1is then determined numerically to compensate

for the deviation of Urfrom 1.0, fnew

1=fold

1+α(1 −Ur), where α > 0 is taken as

a constant of the order of ρr/dt(where dtis the time-step). If Ur<1.0, f1increases

proportionally to increase Urin the next time-step, and vice versa. This method yields

a maximum |Ur(t)−1.0|of 10−5for all time-steps after the dynamically steady state is

reached. The simulations are conducted at Re = 3000 and M= 1.5, assuming Pr = 0.7

and that the dimensionless viscosity (normalized by its wall value µr) and temperature

satisfy µ=T0.7.

The respective numbers of grid points in the x,yand zdirections are nx= 800,

ny= 200 and nz= 400. For the present channel size and Reynolds number, the spatial

resolution yield ∆x+,∆y+

max and ∆z+less than 3.0, which is suﬃciently ﬁne for DNS.

The ﬁrst 3 grid points in the wall-normal direction (where a non-uniform grid is used)

are in the y+<1.0 region. The simulations are run in a suﬃcient amount of simulation

8M. Aghaei-Jouybari et al.

Figure 2. Proﬁles of mean and turbulence variables for the smooth-wall ﬂow at Re = 3000

and M= 1.5: present simulation, Coleman et al. (1995). (a) Mean values of

temperature, streamwise velocity and density, (b) r.m.s. of turbulent velocities (no summation

over Greek indices, ∗denotes normalization in wall units using τw,s =µrdu

dy

wand ρw), (c)

r.m.s. of density, and (d) Reynolds shear stress.

time to reach the steady state. Thereafter the statistics are averaged over approximately

20 large eddy turn over time (δ/uτ,avg , where uτ,av g is an average value of the friction

velocities on both walls, see table 2 for deﬁnition).

2.4. Validation of the numerical method and the IB method

The numerical method is validated by simulating a smooth-wall supersonic turbulent

channel ﬂow at M= 1.5 and Re = 3000. The same setup was employed by Coleman

et al. (1995), which is used here as the benchmark study.

Figure 2 compares mean and turbulence statistics of the present simulation with those

of Coleman et al. (1995). The two simulations are in a good agreement for mean velocity,

density and temperature, as well as for Reynolds stresses and density variance. This

veriﬁes the numerical solver.

To validate the proposed IB method, we simulated case C1 in two ways: one using

the IB method and the other solving the conventional NS equations on a body-ﬁtted

mesh setup. The contour of level set function ψfor the IB method and the mesh of

the conformal setup are compared in ﬁgure 3. The contour line corresponding to ψ= 0

represents well the ﬂuid-solid interface.

Figure 4 compares the results obtained using the IB method and those using a body-

Supersonic ﬂows over rough walls 9

Figure 3. Contour of level set ψ, ranging from -1 (blue) to +1 (yellow) used for the IB method

(a), and mesh used for the conformal setup (b), both for Case C1. In subplot (a) the solid white

and black lines indicate, respectively, the exact roughness height as in equation (2.9) and the

iso-line of ψ= 0 obtained from the level-set equation (2.4); the diﬀerence between the two lines

is one grid point maximum. The inset in (a) zooms in to show the interface.

ﬁtted mesh, in terms of various mean and turbulence variables, including mean proﬁles

of velocity, temperature and density, Reynolds stresses, and variance of temperature. All

plots show a good agreement between the two simulations. One notices, however, slight

diﬀerences (about 5%) in the u′′′ r.m.s. proﬁles near the crest elevation of roughness at

y+= 25. This is probably due to the diﬀerent meshes used in the immersed boundary

simulation and the conformal one and the interpolation scheme used in the conformal case

to convert ﬂuid ﬁelds to the Cartesian coordinates before the statistics were calculated.

Overall, these results validate the use of the present IB method to characterize single-

point statistics of mean ﬂow and turbulence. See Movie 1 for ﬂow visualizations for case

C1 with the IB method.

3. Results

3.1. Visualizations of the instantaneous and averaged ﬁelds

Instantaneous vortical structures are visualized using iso-surfaces of Q-criterion (Chong

et al. 1990) in ﬁgure 5. Modiﬁcations of the near wall turbulence on the rough-wall side

are noticeable. The main diﬀerence between the eﬀects of diﬀerent roughness geometries

is the shock patterns shown by the instantaneous numerical schlieren images (ﬁgure 6).

These patterns are also persistent in time as shown by time- and spanwise-averaged ∇ · u

(ﬁgure 7). Both ﬁgures show that 2D surfaces (cases C1 and C2) induce strong shocks

that reach the upper wall and are reﬂected back to the domain after impingement. The

shock patterns exhibit the same wavelength of the roughness geometries, and inﬂuence

the ﬂow properties in the whole channel. This is obvious in the contours of instantaneous

temperature ﬁelds in ﬁgure 8, where temperature periodically changes in the compression

and expansion regions associated with roughness geometries in C1 and C2. For 3D cases

the embedded shocks are weaker and, consequently, replaced by the small-scale shocklets.

10 M. Aghaei-Jouybari et al.

Figure 4. Mean and turbulence variables for case C1, simulated using the IB method ( )

and the conformal mesh ( ): mean temperature, streamwise velocity and density (a), r.m.s.

of velocity components in plus units (roughness side, b), time and spanwise average of velocity

and temperature at the roughness crest and valley locations (c, crest in blue and valley in

black), and r.m.s. of temperature (d). In (d), note that temperature r.m.s. is theoretically zero

at the roughness trough (y= 0); the intrinsic-averaged value in y≈0 region for the IBM case,

however, ﬂuctuates due to the limited ﬂuid area. This region is removed form the plot. The

vertical dot-dash lines show y=kc. Superscript + denotes normalization in wall units using

uτ,r (tabulated in table 2) and ρr.

Case uτ,s /Uruτ,r /Uruτ,avg /UrReτρw∆U ∗

V D ∆U +

V D Cf×103

C1 0.0629 0.0997 0.0834 250 1.539 7.76 7.72 13.9

C2 0.0634 0.0912 0.0785 236 1.484 6.61 6.64 12.3

C3 0.0642 0.0847 0.0751 225 1.456 5.13 5.15 11.3

C4 0.0644 0.0899 0.0782 235 1.451 6.25 6.3 12.2

SM 0.0633 – 0.0633 190 1.360 – – 8.0

Table 2. Wall friction comparison. uτ,s =pτw,s /ρrand uτ,r =pτw,r /ρr, where

τw,s =−µrd<u>

dy

y=2δis wall shear stress on the smooth side and τw,r =1

AtRVff1dv−τw,s is that

on the rough side (obtained from momentum balance). Reτ=ρruτ,avg δ/µr,Cf= 2(uτ ,avg /Ur)2,

u2

τ,avg =u2

τ,s +u2

τ,r /2, ρwis the density value at y= 0, and ∆U∗

V D and ∆U +

V D are the roughness

functions associated with the Van Driest transformed velocities.

Supersonic ﬂows over rough walls 11

Figure 5. Isosurfaces of Q= 3 (in blue, normalized by Urand δ) for all rough cases. The gray

isosurfaces show the roughness surfaces.

3.2. Mean and turbulence variables

First, the values of frictional velocities on the smooth and rough sides as well as the

frictional Reynolds number Reτand drag coeﬃcient Cfare tabulated in table 2 for all

cases. On the rough-wall side, the wall friction include both viscous and pressure drag

components. Due to diﬀerences in wall friction generated by roughnesses of diﬀerent

geometries, Reτvaries from 190 to 250. Yet, the ﬂows are all low-Reynolds-number ones;

the diﬀerences in shock features and ﬂow statistics (discussed thoroughly below) are thus

likely a result of the change in roughness geometry, instead of the change in friction

Reynolds number.

The comparison shows that, as expected, the wall friction on the rough-wall side is

higher than that on the smooth wall side for all cases. Overall, a 2D roughness generates

higher friction than a 3D one of the same height. This is consistent with observations

in incompressible ﬂows (e.g. by Volino et al. 2011) that 2D roughness aﬀects turbulence

more strongly due to the larger length scale (in z) that is imparted to the ﬂow. In addition,

results show that for 3D roughnesses a higher friction is obtained for a shorter wavelength

(or higher roughness slope), which is also consistent with observations in incompressible

ﬂows (Napoli et al. 2008). Between the two 2D rough surfaces, however, the one with

a higher slope (C2) yields a lower wall friction. As will be shown later in section 3.3,

this appears to be a result of stronger turbulent mixing above the rough surface in C1

than in C2, due to regions with more intense compression (those with strong negative

values of ∇·u, ﬁgure 7) in which strong turbulent-kinetic-energy (TKE) production and

redistribution take place. These observations indicate that, in fully developed supersonic

rough-wall ﬂows, the dependency of the wall friction on roughness geometry is more

complex than in incompressible ﬂows, due to compressibility eﬀects. Future systematic

studies with a wide range of diﬀerent rough surfaces are needed to detail the dependences

of shocks on roughness height and geometry, and analyses of near-wall momentum balance

are needed to further understand changes of the ﬂow.

Figure 9 compares proﬁles of the mean and turbulence quantities between diﬀerent

12 M. Aghaei-Jouybari et al.

Figure 6. Numerical schlieren images, showing contours of instantaneous log|∇ρ|. For a better

visualization the contour ranges are chosen diﬀerently for diﬀerent cases. ∇ρis normalized by

ρrand δ.

cases. The mean streamwise velocity (ﬁgure 9a) and density (not shown) are both weakly

dependent on the roughness geometry across the channel, except for the region near the

rough wall. This is because the normalization using the bulk values (Urand ρr) absorbs

major diﬀerences in the velocity and density proﬁles in the bulk part of the channel.

The mean temperature values (ﬁgure 9c), on the other hand, diﬀer across the channel for

diﬀerent roughness topographies. The mean temperature is higher for 2D roughness cases

(C1 and C2) compared to the 3D ones (C3 and C4) and the smooth case (SM). Here, the

temperature is normalized by the wall value, Tw, which does not absorb the diﬀerences

in the core region. It is established (Anderson 1990, chapter 3) that shock waves result

Supersonic ﬂows over rough walls 13

Figure 7. Contours of ∇ · uaveraged in time and spanwise direction. All normalized by Ur

and δ. To calculate the spanwise-averaged values, intrinsic averaging was performed along the

spanwise direction at each (x, y) point. An xslice of the corresponding rough surface is shown

in each subplot.

in entropy generation, because of strong viscous eﬀects and thermal conduction in large

gradients regions. The stronger shocks in the 2D roughness cases involve more entropy

in the domain than in the 3D cases. As a result, the irreversible heat generation is more

intense for these cases, leading to higher temperature values.

The r.m.s. of the three u′′′

iﬂuctuation components are plotted in ﬁgure 9(b) in wall

units. For rough cases, it shows that roughness eﬀects are mostly conﬁned to a near-wall

region; outside this region the diﬀerences between proﬁles for diﬀerent rough surfaces are

smaller for all velocity components. This is similar to the concept of roughness sublayer,

14 M. Aghaei-Jouybari et al.

Figure 8. Contours of instantaneous T, normalized by Tr.

deﬁned as the near-wall layer where turbulence statistics in wall units vary with the wall

condition (Flack et al. 2007), in an incompressible turbulent ﬂow bounded by rough wall.

Near the wall, the v′′′ and w′′′ components are similar among all cases, whereas the u′′′

components in 3D cases display a peak closer to the wall than their 2D counterparts.

Similar phenomena were observed for incompressible ﬂow; it was explained as a result

of a thicker roughness sublayer over a 2D roughness (Volino et al. 2011), leading to

a peak farther from the smooth-wall peak elevation at y+≈15. The fact that the

turbulence intensities in wall units do not collapse perfectly in the outer layer (i.e. the

region above the roughness sublayer) among all cases indicates that the wall similarity

(or “outer layer similarity”, Schultz & Flack 2007) of Townsend (1976) does not apply.

Supersonic ﬂows over rough walls 15

Figure 9. Mean and turbulence variables for cases C1 ( ), C2 ( ), C3 ( ), C4 ( )

and SM ( ): proﬁles of the double-averaged streamwise velocity (a), r.m.s. of velocities in

plus units (roughness side, b), double-averaged temperature (c), and r.m.s. of temperature (d).

In (d), note that temperature r.m.s. is theoretically zero at the roughness trough (y= 0); the

intrinsic-averaged value in y≈0 region, however, ﬂuctuates due to the limited ﬂuid area. This

region is removed form the plot.

The wall similarity hypothesis (primarily describing incompressible ﬂows) states that, at

high Reynolds number and with very small roughness compared to δ, turbulent statistics

outside the roughness sublayer are independent of wall roughness, except for its scaling

on the friction velocity. Given the relatively low Reynolds number and large roughness

(kc/δ = 0.1) in the present cases, exact wall similarity is not expected. In addition,

in current supersonic ﬂows roughness is shown to directly aﬀect outer layer turbulence

through its eﬀect on shocks that extend to the core region of the channel ﬂow (ﬁgures 6

and 7).

The proﬁles of temperature r.m.s. in ﬁgure 9(d) show that the intensity of temperature

ﬂuctuations far from the wall depends strongly on the roughness geometry. For 2D rough

surfaces, the variations of curve shape in the bulk of the channel are associated with the

shock patterns in the domain. Temperature varies signiﬁcantly near the locations where

the shock waves coincide and form nodes of shock diamonds (i.e. the nodes away from

walls). These shock diamonds are also visible in ﬁgures 6, 7 and 8 (C1 and C2). For 3D

cases the shock diamonds are weak or nonexistent. Therefore, the curves of temperature

r.m.s. in 3D cases are smooth in the core region.

Figure 10 compares the mean velocity proﬁles in inner units. The law of the wall for

mean velocity scaled in this way refers to the universal logarithmic proﬁle in regions

between 50δν≲y≲0.2δ(where δνis the viscous length scale) on a smooth wall.

On a rough wall, incompressible ﬂow studies showed that the logarithmic proﬁle still

16 M. Aghaei-Jouybari et al.

Figure 10. Law of the wall. Proﬁles of mean velocities transformed using (a) original Van Driest

transformation (equation 3.1) and (b) a modiﬁed Van Driest transformation (equation 3.2).

Cases C1 ( ), C2 ( ), C3 ( ), C4 ( ) and SM ( ). Solid magenta lines

( ) shows slope of 1/κ, where κ= 0.41 is the von Karman constant. In subplot (a) results

of Coleman et al. (1995) ( ) and Foysi et al. (2004) ( ) for smooth-wall ﬂows are provided

for comparison, and the blue solid line ( ) is same as the C1 proﬁle in subplot (b). ∆U ∗

V D

and ∆U+

V D are roughness functions for case C1.

exists, with its lower extent shifted to the top of roughness sublayer. Both roughness

and compressibility were found to inﬂuence the law of the wall through their eﬀects on

the inner units. Speciﬁcally, roughness shifts the logarithmic proﬁles downward for an

amount ∆U+(called roughness function) with respect to a smooth-wall ﬂow (Nikuradse

1933). This has been observed for a wide range of roughness topographies (Raupach et al.

1991; Schultz & Flack 2007; Leonardi et al. 2007; Forooghi et al. 2017; Busse et al. 2017;

Womack et al. 2022, to name a few).

For compressible ﬂows, ﬁnding appropriate inner velocity, length, density and viscosity

scales that result in universal law of the wall is an active subject of research for smooth-

wall ﬂows (Morkovin 1962; Volpiani et al. 2020, among many others). The complexities

stem from signiﬁcant variations in density, viscosity and heat transfer across the boundary

layer that need to be accounted for. Here we plot density-transformed mean velocity

proﬁles, introduced by Van Driest (1951). The approach has been shown to collapse

mean velocity proﬁles for smooth wall ﬂows at diﬀerent Mach numbers (Guarini et al.

2000; Lagha et al. 2011; Trettel & Larsson 2016). The original Van Driest transformation

reads as

U∗

V D =Z⟨u⟩∗

0⟨ρ⟩

ρw1/2

d⟨u⟩∗,(3.1)

where superscript ∗denotes normalization using τw,r ,ρw=⟨ρ⟩|y=0 and µr. The results

are plotted in ﬁgure 10(a). The proﬁles of Coleman et al. (1995) and Foysi et al. (2004)

for smooth-wall ﬂows (M= 1.5 and Re = 3000) are also compared. The present SM

proﬁle matches very well with the reference data. We also employ a modiﬁed Van Driest

transformation, where all density scales are normalized by ρrinstead of ρw. The modiﬁed

Van Driest transformation is

U+

V D =Z⟨u⟩+

0⟨ρ⟩

ρr1/2

d⟨u⟩+,(3.2)

Supersonic ﬂows over rough walls 17

where superscript + denotes normalization using τw,r ,ρrand µr. The results are plotted

in ﬁgure 10(b). Since both ρwand ρrare in the order of unity, U∗

V D and U+

V D are not

signiﬁcantly diﬀerent (comparing the solid blue and black lines in ﬁgure 10a for case

C1). A constant displacement height d= 0.8kcis chosen for the rough cases in ﬁgure 10.

All rough-wall proﬁles in ﬁgure 10 show a downward shift (∆UV D) with respect to the

smooth wall due to higher wall friction, similar to incompressible ﬂows. The magnitudes

of ∆U∗

V D and ∆U+

V D are measured at (y−0.8kc)∗= 100 and (y−0.8kc)+= 100,

respectively, and tabulated in table 2. The two diﬀerent transformations give virtually the

same roughness functions, which are larger for 2D rough cases than 3D ones and display

the same comparison as that of Cfamong all cases (table 2). These observations suggest

that the discussions in the literature concerning equilibrium incompressible rough-wall

drag laws may be extendable to equilibrium supersonic rough-wall ﬂows, when the Van

Driest types of transformation are employed, as long as the Mach number is not too

high. The latter may be necessary as essential dynamics of turbulence in equilibrium

compressible ﬂows are expected to remain similar to their incompressible counterparts, if

the Mach number is not high enough to yield prevailing compressibility eﬀects (Morkovin

1962). Also, one notices in ﬁgure 10 that 1/κ (where κ≈0.41 being the von Karman

constant) is still a good approximation for the slopes of both U∗

V D and U+

V D proﬁles for

the present rough cases, though with minor noticeable variations.

3.3. Budgets of the Reynolds stresses

The transport equation for various components of the Reynolds stress tensor in a

compressible ﬂow reads as (Vyas et al. 2019)

∂

∂t (ρu′′

iu′′

j) = Cij +Pij +DM

ij +DT

ij +

DP

ij +Πij +ϵij +Mij ,

(3.3)

where i,j={1, 2, 3}and C,P,DM,DT,DP,Π,ϵand M, are, respectively, mean con-

vection, production, molecular diﬀusion, turbulent diﬀusion, pressure diﬀusion, pressure-

strain, dissipation, and turbulent mass ﬂux terms, and are deﬁned as

Cij =−∂

∂xk

(ρu′′

iu′′

jeuk),

Pij =−ρu′′

iu′′

k

∂euj

∂xk−ρu′′

ju′′

k

∂eui

∂xk

,

DM

ij =∂

∂xk

(u′′

iτkj +u′′

jτki),

DT

ij =−∂

∂xk

(ρu′′

iu′′

ju′′

k),

DP

ij =−∂

∂xk

(p′u′′

iδjk +p′u′′

jδik),

Πij =p′∂u′′

i

∂xj

+∂u′′

j

∂xi,

ϵij =−τki

∂u′′

j

∂xk−τkj

∂u′′

i

∂xk

,

Mij =u′′

i∂τ kj

∂xk−∂p

∂xj+u′′

j∂τ ki

∂xk−∂p

∂xi.

(3.4)

18 M. Aghaei-Jouybari et al.

Figure 11. Budget balances of TKE. All terms are double-averaged in time and in the x-z

plane. They are normalized by the outer units ρr,Urand δfor the external subplots, and wall

(+) units ρr,uτ,r and µrin the insets. The vertical dash lines show y=kc.

Supersonic ﬂows over rough walls 19

Figure 12. Budget balances of B11. All terms are double-averaged in time and the x-zplane.

They are normalized by the outer units ρr,Urand δfor the external subplots, and wall (+)

units ρr,uτ,r and µrin the insets. Subplot P&Πcompares the production and pressure-strain

terms of all cases: C1 ( ), C2 ( ), C3 ( ), C4 ( ) and SM ( ). The vertical

dash lines show y=kc.

20 M. Aghaei-Jouybari et al.

Figure 13. Contours of P11 normalized using ρr,Urand δ. An xslice of the corresponding

rough surface is shown in each subplot.

The budget terms are calculated for all non-zero components of the Reynolds stress

tensor and for TKE. The budget balance of the transport equation of ⟨ρu′′

iu′′

j⟩is denoted

as Bij . Figures 11 and 12 show wall-normal proﬁles of the spatial-averaged budget terms

of TKE and ⟨ρu′′u′′ ⟩, respectively. Both normalizations by the reference units (ρr,Ur

and δ) and by the wall units (ρr,uτ,r at bottom wall and µr) are used. The residual of

the calculated budget balance, σ, is presented; it is less than 1% of the maximum value

of the shear production Pin all cases. This suggests that the budget terms are calculated

correctly and that the numerical dissipation (as a result of both the solver’s ﬂux-splitting

procedure and the IBM) is small for estimation of the budget terms.

Comparing the smooth- and rough-wall cases, the main diﬀerences are seen near the

Supersonic ﬂows over rough walls 21

wall. Speciﬁcally, molecular diﬀusion and viscous dissipation are non-zero on the smooth

wall, whereas they are zero on the bottom of the rough wall attributed to a quiet region

without turbulence at the root of the roughness elements. Overall, both production

and pressure-strain term on a rough wall peak at elevations near the roughness crest,

independent of the smooth-wall peak elevations.

Among the rough cases, the magnitudes of the budget terms normalized by the

reference values are shown to be modiﬁed by the roughness topography, with the 2D

surfaces producing higher magnitudes than the 3D ones. Two important terms, P11 and

Π11 are compared in ﬁgure 12 among all cases. The comparison among the magnitudes

of both P11 and Π11 displays the same trend as those of the temperature proﬁles (ﬁgure

9c) and Cf(table 2), suggesting that an enhancement of turbulence processes augments

temperature and hydrodynamic drag. For further explanations, the contours of P11 in

an (x, y) plane are shown in ﬁgure 13 to compare the spatial distribution of this term. It

shows that 2D roughness elements lead to stronger turbulence production downstream

of each roughness peak than the 3D roughnesses. This is probably because the 2D

roughnesses induce organized recirculation regions that are aligned in zand stronger

shear layers around the recirculation regions. In addition, turbulence production above a

2D roughness is enhanced by the strong mutual interaction between shock waves. In 2D

roughness cases the regions where two oblique shock waves impinge together (ﬁgure 7) are

associated with enhanced turbulence production, whether it is on the rough- or smooth-

wall side.

The eﬀect of shocks on turbulence is an important phenomenon and represents a

fundamental diﬀerence between supersonic and subsonic turbulent ﬂows over rough walls

– for incompressible ﬂows most of the roughness eﬀects are conﬁned to near wall regions

and the outer layer is expected to be independent of the wall condition (except for

the scaling of outer-layer statistics on the friction velocity), also known as outer layer

similarity (Townsend 1976; Raupach et al. 1991; Schultz & Flack 2007). However, for

the supersonic cases herein, the eﬀects of wall roughness propagate across the channel

and modify turbulence production in the upper wall region via the generated oblique

shocks. The same process occurs on the rough-wall side, where the reﬂected shocks

from the smooth-wall side impinge back to the rough wall and enhance the turbulence

production in these regions. In other words, turbulence processes on both walls depend on

the interaction of shocks, which are themselves dependent on the roughness topography.

This indicates that outer layer similarity does not apply to such supersonic channel

ﬂows, at least for the Reynolds number, Mach number and wall roughness in the present

simulations. The far-reaching eﬀect of surface details may be of potential use in ﬂow and

turbulence control.

3.4. Conditional analysis

In this section, contributions from regions of either expansion, compression or

solenoidal ﬂow to the overall TKE production is analyzed and compared among all

cases. The velocity divergence (∇·u) is used as a measure of compressibility. Large

magnitudes of ∇·ucorrespond to regions of strong compression or expansion immediately

before and after a shock wave.

Figures 14 (a-e) show the probability density function (p.d.f.) of ∇·u, evaluated at

each y-location and normalized to yield the maximum value of 1. There Ddenotes values

taken by ∇·u. To calculate the p.d.f. at all yelevations, the bin width is chosen as

a constant 0.2 in reference units, with (∇·u)max = 40, (∇·u)min =−40. For the 2D

roughness cases (C1 and C2), velocity divergence values far from the walls scatter towards

much larger magnitudes than those near the wall, while they remain in the vicinity of zero

22 M. Aghaei-Jouybari et al.

Figure 14. Probability density functions of velocity divergence evaluated at all yvalues (a-e),

conditional expectations of TKE shear production given velocity divergence E(PTK E

∇·u=D)

(f-j), and their respective products (k-o). Subplots (a-e) are normalized to yield the maximum

value of 1.

for the 3D and smooth-wall cases. This is a quantitative comparison showing the more

signiﬁcant compressibility eﬀects in the 2D roughness cases owing to the strong oblique

shock waves away from the walls. It also conﬁrms that the 3D roughnesses do not induce

strong compressibility eﬀects in the ﬂow, similar to what happens in the smooth-wall

case.

Next, the TKE production conditionally averaged based on velocity divergence,

E(PT KE ∇·u=D), evaluated at each y, is shown in ﬁgures 14 (f-j). In all cases, regions

with large magnitudes of Dcontribute signiﬁcantly to PTK E , both near and far from

the wall. This is prominent in the 2D roughness cases and, to a lower extent, in the

3D-roughness and smooth-wall cases. Although high-magnitude Devents are associated

with signiﬁcant PT KE , their probability of occurrence is low according to the p.d.f. of

∇·u. To assess the distribution of the actual amount of PT KE attributed to regions of

Supersonic ﬂows over rough walls 23

Figure 15. Proﬁles of spatially averaged Pii ( ) and ϵii ( ) conditioned on various

types of compressibility. Quantities are normalized using ρr,Urand δ.

diﬀerent ∇·uvalues, the product between p.d.f. of ∇·uand E(PT KE ∇·u=D) is

plotted in ﬁgures 14(k-o), for each Dand at each y. Results show that the majority of

TKE production comes from low-compressibility regions, due to their high probabilities

of occurrence. The 2D roughnesses lead to larger fractions of TKE production from

negative-D(or compression) events.

To quantitatively compare individual contributions from regions with diﬀerent types

of compressibility to the overall TKE production, ﬁgure 15 shows proﬁles of PT KE

and ϵT KE conditioned on three types of events: compression events (where ∇·u

Ur/δ ⩽

−0.15), solenoidal events (−0.15 <∇·u

Ur/δ <0.15) and expansion events ( ∇·u

Ur/δ ⩾0.15).

The threshold value 0.15 is chosen from ﬁgure 14(e) as approximately the maximum

magnitude reached in the smooth case. In other words, for the purpose of the conditional

analysis, the smooth-wall ﬂow at the present Mach number is considered only weakly

compressible (relative to the rough-wall cases). For the 3D-roughnesses and smooth-wall

cases, the solenoidal events are shown responsible for almost all of the TKE production

24 M. Aghaei-Jouybari et al.

(and dissipation rate). However, for 2D surfaces the compression events contribute as

much as 30% on these processes near the wall, while the contribution from expansion

events is about 5%; far from the wall (y/δ > 0.2), the total contribution from compression

events overtakes that from solenoidal ones. This indicates that the shocks, dependent on

the roughness topography, dynamically inﬂuences the turbulent ﬂow across the channel.

For further understanding of how roughness geometry aﬀects the Reynolds stress balance,

future studies are needed to characterize the dilatation terms (i.e. pressure dilatation and

dilatation dissipation, Sarkar et al. 1991), which are normally small at present Mach and

Reynolds numbers in channel ﬂow with smooth walls but may be signiﬁcant on rough

walls due to the presence of shocks.

4. Concluding remarks

Eﬀects of surface roughness and its topography on compressible turbulent ﬂows were

characterized based on simulations of four supersonic channel ﬂows at M= 1.5 and the

bulk Reynolds number Re = 3000 with top smooth walls and four diﬀerent roughness

geometries on the bottom walls. A baseline smooth-wall channel was also simulated. A

modiﬁed level-set/volume-of-ﬂuid immersed boundary method was used to impose the

boundary conditions of velocity and temperature on the surface of roughness. The method

was validated in terms of mean and turbulence statistics against a companion conformal

mesh simulation. The four roughnesses include two 2D and two 3D sinusoidal surfaces.

The surfaces shared the same peak-to-trough height (of 10% channel half height) but

diﬀered in the surface wavelength.

Results showed signiﬁcant modiﬁcations of turbulence across the channel by the

roughness. Roughness generates a distribution of oblique shocks in connection to the

roughness geometry. These shocks are much stronger on the 2D roughnesses; they reach

across the channel, reﬂected back from the smooth-wall side, and interact to form shock

diamonds. Such strong shock patterns are absent in the smooth-wall channel.

The strong shocks generated by the 2D roughnesses lead to stronger irreversible heat

generation and higher temperature values in the bulk of the channel, than in the 3D-

roughness cases. Roughness on one side of the channel enhances TKE production on both

sides. Conditional analyses based on local compressibility showed that, although high-

compressibility regions are associated with signiﬁcantly enhanced local TKE production,

the probability of occurrence of such regions is low and dependent on the roughness

geometry. For the 2D roughness cases up to 30% of TKE production and dissipation

near the rough wall (as well as most of the outer-layer production and dissipation)

occur in compression regions. However, for the 3D-roughness and smooth-wall cases these

processes are almost all associated with relatively solenoidal events.

This work identiﬁes the mechanism through which wall roughness and its texture aﬀect

a compressible turbulence. It demonstrates that in supersonic channels the roughness

eﬀects can propagate throughout the boundary layer. In this case, the outer layer

similarity established for incompressible ﬂows does not apply. This work focuses on ﬂow

statistics. To fully characterize roughness eﬀects in compressible ﬂows, future studies on

spectral and structural eﬀects, scaling and variable density eﬀects are needed.

Acknowledgements

Computational support was provided by Michigan State University’s Institute for

Cyber-Enabled Research.

Supersonic ﬂows over rough walls 25

Declaration of Interests

The authors report no conﬂict of interest.

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