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Supersonic turbulent flows over sinusoidal rough walls



Direct numerical simulations were performed to characterize fully developed supersonic turbulent channel flows over isothermal rough walls. The effect of roughness was incorporated 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.
This draft was prepared using the LaTeX style file belonging to the Journal of Fluid Mechanics 1
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:
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
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
∂t +
∂t +
∂xjρuiuj+ij 1
Reτij =f1δi1,(2.1b)
∂t +
Reujτij +1
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/(γ1) + ρuiui/2 is the total energy, γCp/Cvis the ratio of specific heats
(assumed to be 1.4), τij =µ∂ui
∂xkδij is the viscous stress tensor, and
∂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 ρrUrLrr,MUr/γRTr, and Pr µrCpc, 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=CpCv. 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
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,
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
∂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,
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.,
θϕθ + (1 ϕ)θb,(2.6)
for Dirichlet BC and
∂n =θ·b
∂n b
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
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
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
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
k(x, z) = 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) = 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(kkav g )2dA is the
root-mean-square (r.m.s.) of roughness height fluctuation, Ra=1
AtRx,z |kkavg |dA is
the first-order moment of height fluctuations, Exi=1
dA is the effective
slope in the xidirection, Sk=1
AtRx,z(kkav g )3dA.k3
rms is the height skewness, and
AtRx,z(kkav 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
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
VfRVfρdvand Ur1
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+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 105for 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
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 y0 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 rand uτ,r =pτw,r r, where
τw,s =µrd<u>
y=2δis wall shear stress on the smooth side and τw,r =1
AtRVff1dvτw,s is that
on the rough side (obtained from momentum balance). Reτ=ρruτ,avg δ/µr,Cf= 2(uτ ,avg /Ur)2,
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 y0 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 = 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 50δνy0.2δ(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, 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
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
V D =Zu
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
V D =Zu+
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 (y0.8kc)= 100 and (y0.8kc)+= 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 (where κ0.41 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′′
j) = Cij +Pij +DM
ij +DT
ij +
ij +Πij +ϵij +Mij ,
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 =
Pij =ρu′′
ij =
iτkj +u′′
ij =
ij =
iδjk +pu′′
Πij =p∂u′′
ϵij =τki
Mij =u′′
i∂τ kj
j∂τ ki
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′′
jis 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
(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
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
0.15), solenoidal events (0.15 <·u
Ur <0.15) and expansion events ( ·u
Ur 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.
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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In the present study, we investigate the compressibility effects in supersonic and hypersonic turbulent boundary layers under the influence of wall disturbances by exploiting direct numerical simulation databases at Mach numbers up to 6. Such wall disturbances enforce extra Reynolds shear stress on the wall and induce mean streamline curvature in rough wall turbulence that leads to the intensification of turbulent motions in the outer region. The turbulent and fluctuating Mach numbers, the density and the velocity divergence fluctuation intensities suggest that the compressibility effects are enhanced by the increment of the free-stream Mach number and the implementation of the wall disturbances. The differences between the Reynolds and Favre average due to the density fluctuations constitute approximately $9\,\%$ of the mean velocity close to the wall and $30\,\%$ of the Reynolds stress near the edge of the boundary layer, indicating their non-negligibility in turbulent modelling strategies. The comparatively strong compressive events behaving as eddy shocklets are observed at the free-stream Mach number of $6$ only in the cases with wall disturbances. By further splitting the velocity into the solenoidal and dilatational components with the Helmholtz decomposition, we found that the dilatational motions are organized as travelling wave packets in the wall-parallel planes close to the wall and as forward inclined structures in the form of radiated waves in the vertical planes. Despite their increased magnitudes and higher portion in the Reynolds normal and shear stresses, the dilatational motions show no tendency of contributing significantly to the skin friction and the production of turbulent kinetic energy due to their mitigation by the cross-correlation between the solenoidal and dilatational velocity components.
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In the present study, we perform direct numerical simulations to investigate the spatial development and basic flow statistics in the supersonic turbulent boundary layers at the free-stream Mach number of 2.0 over smooth and disturbed walls, the latter of which enforce extra Reynolds shear stress in the streamwise direction to emulate the drag increment and mean streamline curvature effects of rough walls. Such disturbances escalate the growth rate of turbulent boundary layer thickness and the shape factor. It is found that under the rescaled global coordinate, the mean velocity, Reynolds stresses and pressure fluctuation variances manifest outer-layer similarity, whereas the average and fluctuation variances of temperature and density do not share such a property. Compressibility effects are enhanced by the wall disturbances, yet not sufficiently strong to directly impact the turbulent kinetic energy transport under the presently considered flow parameters. The generalized Reynolds analogy that relates the mean velocity and temperature can be satisfied by incorporating the refinement in modifying the generalized recovery coefficient, and that associates the fluctuating velocity and temperature works reasonably well, indicating the passive transport of temperature fluctuations. The dispersive motions are dominant and decay exponentially below the equivalent sand grain roughness height ks, above which the wall disturbances are distorted to form unsteady motions responsible for the intensified density and pressure fluctuations in the free-stream travelling isentropically as the acoustic radiations.
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In the present study, we perform direct numerical simulations to study the influences of the spanwise-oriented grooves, which are emulated by the reasonably designed “relaxed” boundary conditions, on the kinetic and thermodynamic statistics in a supersonic turbulent channel flow at the Mach number of 1.5 and Reynolds number of 3000. The phase averaged flow fields show that the relaxed boundary induces compressive and expansive waves that travel across the whole channel and are reflected by the upper wall. These waves are isentropic in the average sense except in the viscous sublayer. In the near-wall region, vortices and streaks that constitute the self-sustaining cycles are less populated and less meandering, while in the outer region, especially near the channel center, the velocity divergence is as strong as the vorticity. The temperature, density, and pressure fluctuations are enhanced by these waves. The correlations between the velocity and temperature are altered, due to the counter effects caused by the vortical motions and isentropic waves.
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We perform direct numerical simulation of supersonic turbulent channel flow over cubical roughness elements, spanning bulk Mach numbers $M_b=0.3$ – $4$ , both in the transitional and fully rough regime. We propose a novel definition of roughness Reynolds number which is able to account for the viscosity variations at the roughness crest and should be used to compare rough-wall flows across different Mach numbers. As in the incompressible flow regime, the mean velocity profile shows a downward shift with respect to the baseline smooth wall cases, however, the magnitude of this velocity deficit is largely affected by the Mach number. Compressibility transformations are able to account for this effect, and data show a very good agreement with the incompressible fully rough asymptote, when the relevant roughness Reynolds number is used. Velocity statistics present outer layer similarity with the equivalent smooth wall cases, however, this does not hold for the thermal field, which is substantially affected by the roughness, even in the channel core. We show that this is a direct consequence of the quadratic temperature–velocity relation which is also valid for rough walls. Analysis of the heat transfer shows that the relative drag increase is always larger than the relative heat transfer enhancement, however, increasing the Mach number brings data closer to the Reynolds analogy line due to the rising relevance of the aerodynamic heating.
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Aiming to study the rough-wall turbulent boundary layer structure over differently arranged roughness elements, an experimental study was conducted on flows with regular and random roughness. Varying planform densities of truncated cone roughness elements in a square staggered pattern were investigated. The same planform densities were also investigated in random arrangements. Velocity statistics were measured via two-component laser Doppler velocimetry and stereoscopic particle image velocimetry. Friction velocity, thickness, roughness length and zero-plane displacement, determined from spatially averaged flow statistics, showed only minor differences between the regular and random arrangements at the same density. Recent a priori morphometric and statistical drag prediction methods were evaluated against experimentally determined roughness length. Observed differences between regular and random surface flow parameters were due to the presence of secondary flows which manifest as high-momentum pathways and low-momentum pathways in the streamwise velocity. Contrary to expectation, these secondary flows were present over the random surfaces and not discernible over the regular surfaces. Previously identified streamwise-coherent spanwise roughness heterogeneity does not seem to be present, suggesting that such roughness heterogeneity is not necessary to sustain secondary flows. Evidence suggests that the observed secondary flows were initiated at the front edge of the roughness and sustained over irregular roughness. Due to the secondary flows, local turbulent boundary layer profiles do not scale with local wall shear stress but appear to scale with local turbulent shear stress above the roughness canopy. Additionally, quadrant analysis shows distinct changes in the populations of ejection and sweep events.
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The dynamical effects of roughness geometry on the response of a half-height turbulent channel flow to an impulse acceleration are investigated using direct numerical simulations. Two rough surfaces different in the surface height spectrum are compared between themselves and with a smooth-wall baseline case. Both rough cases develop from a transitionally rough state to a fully rough one. Results show that on rough walls the thickness of the roughness sublayer (RSL), defined as the layer with significant form-induced stresses, stays almost constant. The ensemble-average flows inside the RSL stays close to equilibrium throughout the transient. This is shown by the form-induced perturbations largely scaling with the mean velocity at the edge of the RSL. Inside the RSL, turbulence develops rapidly to the new steady state, accompanied by substantial changes in the Reynolds stress balance. In contrast, the flow above the RSL recovers long after the sublayer is fully developed, without a significant change in Reynolds stress balance. The geometry of the roughness plays an important role in determining the rate of response of turbulence throughout the boundary layer. This work provides detailed explanation of the suppression of reverse transition by surface roughness in response to a mean flow acceleration.
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Direct numerical simulation (DNS) of flow in a turbulent channel with a random-rough bottom wall is performed at friction Reynolds number $Re_{\tau}=400$ and $600$. The rough surface corresponds to the experiments of Flack et al. (2019). The computed skin friction coefficients and the roughness functions show good agreement with experimental results. The double-averaging methodology is used to investigate mean velocity, Reynolds stresses, dispersive flux, and mean momentum balance. The roll-up of the shear layer on the roughness crests is identified as a primary contributor to the wall-normal momentum transfer. The mean-square pressure fluctuations increase in the roughness layer and collapse onto their smooth-wall levels away from the wall. The dominant source terms in the pressure Poisson equation are examined. The rapid term shows that high pressure fluctuations observed in front of and above the roughness elements are mainly due to the attached shear layer formed upstream of the protrusions. The contribution of the slow term is relatively smaller. The slow term is primarily increased in the troughs and in front of the roughness elements, corresponding to the occurrence of quasi-streamwise vortices and secondary vortical structures. The mean wall shear on the rough surface is highly correlated with the roughness height, and depends on the local roughness topography. The p.d.f of wall-shear stress fluctuations are consistent with higher velocities at roughness crests and reverse flow in the valley regions. Extreme events are more probable due to the roughness. Events with comparable magnitudes of the streamwise and spanwise wall-shear stress occur more frequently, corresponding to a more isotropic vorticity field in the roughness layer.
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This paper investigates a long-standing question about the effect of surface roughness on turbulent flow: what is the equivalent roughness sand-grain height for a given roughness topography? Deep Neural Network (DNN) and Gaussian Process Regression (GPR) machine learning approaches are used to develop a high-fidelity prediction approach of the Nikuradse equivalent sand-grain height k s for turbulent flows over a wide variety of different rough surfaces. To this end, 45 surface geometries were generated and the flow over them simulated at Re τ = 1000 using direct numerical simulations. These surface geometries differed significantly in moments of surface height fluctuations, effective slope, average inclination, porosity and degree of randomness. Thirty of these surfaces were considered fully-rough and they were supplemented with experimental data for fully-rough flows over 15 more surfaces available from previous studies. The DNN and GPR methods predicted k s with an average error of less than 10% and a maximum error of less than 30%, which appears to be significantly more accurate than existing prediction formulas. They also identified the surface porosity and the effective slope of roughness in the spanwise direction as important factors in drag prediction.
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A scaling formula for the mean velocity and wall distance in compressible turbulent wall-bounded flows is developed. The development is both physics- and data-driven: universality in the viscous sublayer and of the Morkovin-scaled shear stress is analytically enforced, after which the remaining two free parameters are determined from direct numerical simulation data. The scaling formula is calibrated on four boundary layers up to Mach 6, and validated on seven boundary layers up to Mach 14. The proposed transformation shows an improved collapse of the mean velocity profile across different Reynolds numbers, Mach numbers, and wall thermal conditions when compared to existing ones.
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The effects of bed roughness, isolated from that of bed permeability, on the vertical transport processes across the sediment-water interface (SWI) are not well understood. We compare the statistics and structure of the mean flow and turbulence in open-channel flows with a friction Reynolds number of 395 and a permeability Reynolds number of 2.6 over sediments with either regular or random grain packing at the SWI. The regular sediment interface is formed by cubic packing of spheres aligned with the mean-velocity direction. It is shown that, even in the absence of any bedform, the subtle details of the particle roughness alone can significantly affect the dynamics of turbulence and the time-mean flow. Such effects translate to large differences in penetration depths, apparent permeabilities, vertical mass fluxes and subsurface flow paths of passive scalars. The less organized distribution of mean recirculation regions near the interface with a random packing leads to a more isotropic form-induced stress tensor. The augmented wall-normal form-induced fluctuations play a significant role in increasing mixing and wall-normal mass and momentum exchange.
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A long-standing problem in turbulence modeling is that the Reynolds stress tensor alone is not necessarily sufficient to characterize the transient and non-equilibrium behaviors of turbulence under arbitrary mean deformation or frame rotation. A more complete single-point characterization of the flow can be obtained using the structure Dimensionality, Circulicity, and Inhomogeneity tensors. These tensors are one-point correlations of local stream vector gradients and carry non-local information regarding the structure of the flow field. We explore the potential of these tensors to improve understanding of complex turbulent flows using direct numerical simulation of flows in channels with a smooth wall and a two-dimensional sinusoidal wavy wall. To enforce no-slip and no-penetration conditions at wavy-wall boundaries, an immersed boundary method for the stream vector Poisson equation was adopted within the framework of F. S. Stylianou, R. Pecnik, and S. C. Kassinos, \textit{Comput. Fluids}, 106, 54--66 (2015). Results show that the effects of wall waviness on the inclination and aspect ratio of the two-point velocity correlation near the wall are reproduced qualitatively by their corresponding single-point tensor representations. In the outer layer, good quantitative agreement is achieved for both parameters. Additional observations on the structural changes of turbulence due to wall waviness and their relevance to turbulence modeling with surface roughness are discussed. The findings of this investigation suggest that single-point structure tensors can be appended to the modeling basis for inhomogeneous flows with geometrically complex boundaries, such as rough-wall flows, to develop improved turbulence models.
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Turbulence structure resulting from multi-fluid or multi-species, variable-density isotropic turbulence interaction with a Mach 2 shock is studied using turbulence-resolving shock-capturing simulations and Eulerian (grid) and Lagrangian (particle) methods. The complex roles that density plays in the modification of turbulence by the shock wave are identified. Statistical analyses of the velocity gradient tensor (VGT) show that density variations significantly change the turbulence structure and flow topology. Specifically, a stronger symmetrization of the joint probability density function (PDF) of second and third invariants of the anisotropic VGT, PDF $(Q^{\ast },R^{\ast })$ , as well as the PDF of the vortex stretching contribution to the enstrophy equation, are observed in the multi-species case. Furthermore, subsequent to the interaction with the shock, turbulent statistics also acquire a differential distribution in regions having different densities. This results in a nearly symmetric PDF $(Q^{\ast },R^{\ast })$ in heavy-fluid regions, while the light-fluid regions retain the characteristic tear-drop shape. To understand this behaviour and the return to ‘standard’ turbulence structure as the flow evolves away from the shock, Lagrangian dynamics of the VGT and its invariants is studied by considering particle residence times and conditional particle variables in different flow regions. The pressure Hessian contributions to the VGT invariants transport equations are shown to be not only affected by the shock wave, but also by the density in the multi-fluid case, making them critically important to the flow dynamics and turbulence structure.
The force partitioning method [Menon and Mittal, J. Fluid Mech. 918, R3 (2021)0022-112010.1017/jfm.2021.359] is employed to decompose and analyze the pressure-induced drag for turbulent flow over rough walls. The pressure drag force imposed by the rotation-dominated vortical regions (Q>0, where Q is the second invariant of the velocity gradient tensor) and strain-dominated regions (Q<0) are quantified using a geometry dependent auxiliary potential field (denoted by φ). The analysis is performed on data from direct numerical simulations (DNSs) of turbulent channel flows, at frictional Reynolds number of Reτ=500, with cube and sand-grain roughened bottom walls. Results from both simulations indicate that the Q-induced pressure drag is the largest contributor (more than 50%) to the total drag on the rough walls. Data are further analyzed to quantify the effects of time-mean (coherent) and incoherent turbulent flow on the Q-induced drag force, and to discuss possible effects of roller and U-shaped structures expected to occur at the crest and midcrest locations of the roughness elements, respectively. Based on the observation that the φ field encodes information about the surface geometry that directly impacts the drag, we provide initial evidence that it can also be used to parametrize the surface drag. Specifically, we propose and test three norms based on the φ field (two of them related to the surface-induced potential flow), and explore the characterization of the Nikuradse equivalent sand-grain height ks based on these parameters for a number of channel flows with different roughness topologies. Data are provided from a suite of DNS cases by Aghaei-Jouybari et al. [J. Fluid Mech. 912, A8 (2021)0022-112010.1017/jfm.2020.1085]. An empirical correlation depending on these φ-based parameters, with five empirically tuned coefficients, is shown to predict ks with average and maximum errors of 10.5 and 26 percent, respectively. The results confirm that a purely geometric quantity, the φ field, provides useful additional information that can be used in drag law formulations.