Ali Amarloo, Pourya Forooghi, Mahdi Abkar

Department of Mechanical and Production Engineering, Aarhus University, Aarhus C, 80 0 0, Denmark

Keywords:Secondary flows Roughness heterogeneity Unstable/convective boundary layer Large-eddy simulation

ABSTRACT Large-scale secondary motions are known to occur in turbulent flows over surfaces with spanwise rough- ness heterogeneity.Numerical studies often use adjacent high- and low-roughness longitudinal strips to investigate these secondary rolls in boundary layers without any thermal stratification.In the present study, the effect of unstable thermal stratification on secondary rolls in a very high-Reynolds-number turbulent flow with spanwise-heterogeneous roughness is investigated by means of large-eddy simula- tion.The strength of the unstable stratification is systematically changed fromL/h= ?20 toL/h= ?1 , whereLandhare Monin-Obukhov length and boundary-layer height, respectively.This range covers the transition from neutral stratification to unstable stratification.The results show that the positive buoy- ancy associated with the unstable thermal stratification acts against the roughness-induced secondary rolls.In the case of unstable stratification, secondary rolls are completely canceled out by buoyancy and replaced by new stronger convection-induced rolls rotating in opposite directions.

The study of turbulent flow over a surface with heterogeneous roughness can be useful in a variety of applications.A prominent example is the interaction of the atmospheric boundary layer (ABL) with the surface of earth containing different scales of rough- ness such as sand, vegetation, and rivers.The effect of heteroge- neous roughness on turbulent boundary layer is widely studied with the assumption of either streamwise- or spanwise-oriented roughness patches.The streamwise-heterogeneous roughness has been broadly investigated in the past (e.g.Ref.[1]).In the present paper, we focus on the spanwise variation of roughness.The re- lated literature is briefly reviewed in the following.

It is well established in the literature that a spanwise hetero- geneity of wall conditions can induce secondary rolls and large- scale heterogeneity in the mean flow (see Refs.[2-4]).Willingham et al.[5] used large-eddy simulation (LES) to show the existence of roughness-induced secondary flows in the ABL.Also, they found that the width of the roughness strips has a considerable effect on the secondary-flow patterns.Anderson et al.[6] did both nu- merical and experimental study of these secondary motions and showed that they are of Prandtl’s secondary flows of the second kind, which are caused by the spatial gradients in the Reynolds stress components.

The effect of spanwise-heterogeneous roughness is generally studied in one of the two categories: “ridge-type” and “strip- type” roughness.In the ridge-type category, the roughness varia- tions are realized by introducing physical height elevations on the wall.Vanderwel and Ganapathisubramani [7] experimentally in- vestigated the secondary motions by systematically changing the spanwise spacing of roughness strips.They showed that secondary flows are appearing when the spacing is roughly proportional to the boundary-layer thickness and upward movement of flow hap- pens over the high-roughness part of the surface.There have been more studies about the effect of ridge-type roughness in turbulent boundary layer by means of experimental set-ups [8,9] , LESs [10] , and direct numerical simulations (DNS) [11-14]).

In the strip-type category of researches about spanwise- heterogeneous roughness, the strips of roughness are present with- out a considerable elevation in height.Willingham et al.[5] studied the effect of two roughness parameters, including the ratio of the high-roughness length to the low-roughness length and the width of the high-roughness strips, in a high-Reynolds-number bound- ary layer using LES.They reported the appearance of large-scale secondary flows induced by roughness heterogeneity but with a downward movement over the high-roughness strips.Chung et al.[15] used DNS in a turbulent boundary layer with a lower Reynolds number and also identified secondary flows induced by roughness heterogeneity and the downward movement of flow over the high- roughness strips.It should be noted that strip-type roughness can be realistic in cases with a very large value for Reynolds number that the roughness height is much smaller than the boundary layer [16] .

In the context of moderate-Reynolds-number flows, Stroh et al.[17] applied no-slip/free-slip strips at the bottom-wall to inves- tigate secondary motions by DNS.They found that the width of strips is highly important for vortex formation.With a low width of strips, an upward flow happened over the no-slip strip; then by increasing the width, the direction changed from upward to down- ward in the case with high width of strips.In another study by Stroh and co-workers [18] , they applied roughness strips condition resolved by an immersed boundary method instead of a no-slip condition, and they identified a change from strip-type to ridge- type behavior by increasing roughness-strip elevation.Zhang et al.[19] used constrained LES for simulation of turbulent flow with spanwise cosinusoidal variation in roughness.They also identified the large secondary rolls with a downward movement of flow over the high-roughness strips.

Although the above-mentioned studies provide a broad vision of secondary motions induced by spanwise heterogeneous rough- ness in neutrally stratified boundary layers, the thermal stratifi- cation and related buoyancy effects are inherent to many real- world flows, ABL being a prominent example.During the night- time, a stable thermal stratification happens inside the ABL and during the daytime, unstable/convective thermal stratification is expected.Both stable and convective boundary layers have been studied in many previous works (see Ref.[20]), but little atten- tion has been paid to the secondary motions induced by roughness spanwise-heterogeneity in thermally stratified turbulent boundary layers.

To the best of our knowledge, in the context of thermally strat- ified turbulent boundary layers with spanwise roughness hetero- geneity, there is only one single effort by Forooghi et al.[21] , in which stably stratified boundary-layer flow was investigated by mean of LES.Their results showed that stable stratification sup- presses the vertical motions and decreases the strength of the first off-wall secondary vortex.There is no report on the effect of un- stable/convective thermally stratified boundary layers.The focus of this study will be the effect of thermally convective/unstable stratification on the secondary motions induced by the spanwise- heterogeneous roughness in the high-Reynolds-number turbulent boundary layers.

For the LES of the flow, an in-house pseudo-spectral solver is utilized.This program has been frequently used in the past for the thermally stratified boundary-layer simulations [21-24] .In this code, the following filtered equations including incom- pressible continuity, incompressible Navier-Stokes, and potential- temperature transport equations are solved in a half channel,

wherei= 1, 2, 3 are streamwise (x), spanwise (y), and wall-normal (z) directions, respectively,is the filtered velocity,tis time,ρis the fluid density,is the filtered pressure,νis the kinematic vis- cosity,is the subgrid-scale (SGS) stress tensor,δijis the Kronecker delta tensor,gis the gravitational acceleration,is the filtered potential temperature,θ0is the reference tempera- ture, the angle brackets represent a horizontal average,is a constant pressure gradient source term, whereuτis the to- tal/bulk friction velocity, andhis the half-channel height, D is the thermal diffusivity,is the SGS heat flux, andSθ=uτθτ/his a thermal source term, whereθτis surface tempera- ture scale.The SGS stress tensor and SGS heat flux are modeled by the anisotropic minimum dissipation (AMD) model including the buoyancy effects [24] .

The computational domain is a rectangular cube with the size ofLx×Ly×Lz= 4 π×2 π×1(h)following Refs.[6,21] .All four side surfaces (in the streamwise and spanwise directions) have periodic boundary conditions, the top surface has zero-stress, zero-flux, and zero vertical velocity boundary conditions [23,25] .The bottom wall has the roughness ofzo,L/h= 10?5except for two strips of high roughnesszo,H/h= 10?3.The two strips are separated by the dis- tance ofsy=π(h)and have width ofls= 0.64(h).At the bottom wall, the shear stresses and heat flux are calculated by the follow- ing equations (see Refs.[26-28]),

whereτw,x,τw,yare wall shear stresses in streamwise and span- wise directions, respectively, theqw,zis the wall heat flux,κis the von Kármán constant,,Δ,,Δ, andare velocity com- ponents and potential temperature at distance ofΔz, respectively,Δzis the height of first off-wall grid in domain,z0is the rough- ness length scale,is the wall-parallel velocity,Lis Moning-Obukhov (MO) length,ψMandψHare corrections for thermal stratification (see Ref.[29]),θsis the surface temperature.

Other parameters considered in our LESs are= 0.4 m/s ,h= 500 m ,ν= 1.5 ×10?5 m2/s ,Pr= 0.7 ,zo,H= 0.5 m ,zo,L= 0.005 m , andθs= 293 K .The friction Reynolds number isReτ= 1.3 ×107; therefore, the flow is not under the influence of the Reynolds number and is called a nominally infinite Reynolds number (e.g., Refs.[10,30]).More details about the solver can be found in Refs.[31,32] .

The convective thermal stratification strength is changed by varying the magnitude of thermal source term (Sθ) in Eq.(3) [21,32-34] .For this purpose, the bulk MO length is varied betweenL/h= ?20 andL/h= ?1 , which are corresponding for an almost neutrally stratified and an unstably stratified case, respec- tively; the cases and their assigned names are listed in Table 1 .For all cases in this paper, the flow statistics are averaged over 50 flow-through times (one flow-through time =Lx/ub, andubis the bulk velocity) after reaching a statistically stationary state.For bet- ter observation of the converged mean flow field, the results are averaged over the converged LESs with different initial conditions.Since the mean flow is repeated twice in they?direction, the re- sults are also averaged over two sides of the domain; therefore, the results are presented on half of the domain (0 ≤y/h≤π)in they?zplane.

Table 1 Convective thermal stratification conditions of LESs with as- signed name.Lis Monin-Obukhov length, andhis half- channel height.

As mentioned before, the existence of high- and low-roughness strips can induce secondary rolls in the mean flow [5,15] , whose topology and strength can both be affected by the thermal strati- fication, at least for a stably stratified boundary layer [21] .In or- der to observe the effect of convective stratification on these sec- ondary motions, Fig.1 shows the vectors of in-plane motions and the contour of normalized streamwise velocity deficit from the present LES.As defined in Eq.(6) , streamwise velocity deficitu＇＇ is the difference between the streamwise- and time-averaged ve- locity and its spanwise averaged value.In Fig.1 , the streamwise velocity deficit is normalized byu0, which is the horizontal aver- age of mean streamwise velocity atz/h= 1 .

Fig. 1. Contours of the normalized streamwise velocity deficit and vectors of the in-plane motion.The thick solid black line indicates the strip with high roughness.Vectors of all cases are on the same scale.(a) Case L-20, (b) Case L-4, (c) Case L-2, (d) Case L-1 .

Here, the overbar indicates the averaged value in time and inx-direction, and the angle bracket indicates the average value iny-direction.

In case L-20 (Fig.1 a), which corresponds for a very low ther- mal stratification, the existence of high and low momentum paths (a.k.a., LMPs and HMPs [2]) and secondary rolls is compatible with the neutral cases in Ref.[5] (Case B), and Ref.[21] (Case N).The comparison of absolute values of velocity deficit shows that higher convective thermal stratification weakens the strength of men- tioned HMP and LMP inside the domain up to a point (at approx- imatelyL/h= ?2) that heterogeneity of streamwise velocity van- ishes.As convective stratification is further intensified (L/h= ?1), new HMP and LMP are induced into the domain.It should be noted that the position of HMP is replaced by LMP, and HMP forms above the low-roughness strip instead of the high-roughness one.There is only one off-wall secondary roll in all cases which gets weaker by higher convective stratification, which disappears ap- proximately atL/h= ?2 .In case L-1 (L/h= ?1), the secondary rolls are replaced by opposite rotating rolls.In other words, the down- ward motion of fluid over the high-roughness strip in the neutral case is replaced by an upward motion in the unstably stratified case.

For a better understanding of the convective stratification effect on the secondary motions of the mean flow, the transport equation of mean vorticity can be derived by applying the curl operation on Reynolds-averaged momentum equation.In Eq.(7) , the curl opera- tion is applied on Eq.(2) , after applying the Reynolds decomposi- tion

The left hand side of Eq.(8) represents the advection of mean streamwise vorticity; on the right hand side, the first curly bracket represents a source term associated with spatial gradients of Reynolds stresses, and the second curly bracket represents a source term associated with buoyancy force.In Fig.2 , contours of these two source terms normalized by(uτ/h)2and the values of stream- wise mean vortex normalized byuτ/hare depicted for all cases.

In the case L-20, Fig.2 a shows that spanwise heterogeneity of wall roughness induces both negative and positive sources for vor- tex advection produced by gradients of Reynolds stresses.Fig.2 b shows that in this case, the buoyancy effect is negligible, therefore the vorticity (depicted in Fig.2 c) is only associated with spatial gradients of Reynolds stresses.The values of vorticity show that even though there are both negative and positive sources on both sides of the high-roughness strip, the negative source is dominant on the left side and the positive source is dominant on the right side; hence, the secondary rolls make a downward motion over the high-roughness strip.

Figure 2 e and 2h shows that the vorticity source term from the buoyancy force gets stronger by a higher convective stratifi- cation.This source term is only positive on the left side of the high-roughness strip and only negative on the right side of it.In this way, the dominance of the negative source associated with Reynolds stress on the left side gets canceled by positive source term from the buoyancy force; hence, the main vortices (i.e.sec- ondary rolls) vanish in the case L-2 (Fig.2 i).

In the case L-1, Fig.2 k shows that the source term due to buoy- ancy force is much higher in comparison to cases L-2 and L-4.The dominance of buoyancy source term results in positive vorticity values on the left side of the high-roughness patch (seen in Fig.2 l) and negative vorticity on the right side of it; hence, the new sec- ondary rolls are rotating in opposite direction, and they make an upward movement over the high-roughness strip.

Fig. 2. Contours of two source terms in transport equation of streamwise mean vorticity (Eq.(8)) and streamwise mean vorticity (a) (b) (c) Case L-20, (d) (e) (f) Case L-4, (g) (h) (i) Case L-2, (j) (k) (l) Case L-1.The source term by spatial gradients of Reynolds stresses (a) (d) (g) (j) and the source term by buoyancy force (b) (e) (h) (k) are normalized by(u τ/h )2 .The streamwise mean voriticy (c) (f) (i) (l) is normalized byu τ/h.

Fig. 3. (a) Normalized mean spanwise-averaged velocity profiles.(b) Normalized mean spanwise-averaged potential temperature profiles.

Figure 3 shows the profile of time- and spanwise-averaged ve- locity normalized by the friction velocity,uτ, and potential tem- perature normalized by the surface temperature scale,θτ.Simi- lar trends are visible for both the velocity and the temperature profiles by changing the MO length.By increasing the convective- stratification intensity, there is a decrease in values of mean veloc- ity and potential temperature (in plus units) which is an expected result of stronger convection in the domain.

To study the formation of LMP and HMP more quantitatively, the root mean square (RMS) of the streamwise velocity deficit and potential temperature deficit (defined in Eq.(6)) are plotted in Fig.4 .Similar to the Fig.3 , the RMSs of deficit values of stream- wise velocity and potential temperature show similar trends by changing the MO length.In the case L-20, there are local mini- mum and local maximum in RMSs, which are related to the ex- istence of HMP over the high-roughness strip and LMP over the low-roughness patch (Fig.1 a).By higher strength for convective stratification (the cases L-4 and L-2), these local extrema tend to fade away.As convective stratification is further increased (the un- stable case L-1), the RMSs of deficit terms are increased again but without any local extrema.

Fig. 4. (a) Normalized mean spanwise-averaged streamwise velocity deficit profiles.(b) Normalized mean spanwise-averaged potential deficit temperature profiles.

Fig. 5. (a) Normalized mean spanwise-averaged wall-normal velocity deficit profiles.(b) Strength of the secondary rolls measured by maximum of wall-normal dispersive stress normalized by that of the neutral case: circles for convective stratification and squares for stable stratification [21] .

Figure 5 a shows the RMS of time- and spanwise-averaged wall- normal velocity deficit normalized by the friction velocity.The sig- nificance of these profiles is in that their peak value and peak loca- tion are an indication of the strength and the center location of the secondary rolls, respectively [21] .It is shown that the center point of secondary rolls doesn’t change significantly at lower values of convective stratification, but the peak values decreases which is the sign of convective stratification acting against secondary rolls created by roughness heterogeneity.When the convective stratifi- cation is further increased (unstable case L-1), new rolls with even stronger intensity and a different center point location appear.This inherent difference is because these new rolls are created by the dominance of convective stratification (i.e.source term by buoy- ancy force) over the roughness-induced sources (i.e.source term by spatial gradients of Reynolds stresses) in streamwise mean vor- ticity transport equation.

To better observe the strength of secondary flows, the peak val- ues from Fig.5 a are plotted againsth/Lin Fig.5 b.To complement the present results, similar quantities for stable cases are readapted from Ref.[21] and plotted in the same figure.The circles indicate the normalized peak values from Fig.5 a and the squares indicate similar quantities for the cases with stable thermally stratification (readapted from Ref.[21]).The strength values of secondary flows are normalized by that of the neutral case re-adapted from Ref.[21] .Forooghi et al.[21] stated that the stable stratification sup- pressed the vertical motions and weakened the strength of sec- ondary flows.In cases with convective/unstable stratification, the strength of secondary flows also decreases, and at a certain point (aroundh/L= ?0.5), the secondary rolls are disappeared.As the convective stratification is further intensified, a new regime with a much stronger convective secondary roll is appeared (h/L= ?1).

In Fig.6 , the profiles of dispersive wall-normal momentum and heat fluxes are shown.For momentum flux, the dispersive stress component 〈u＇＇w＇＇〉 is normalized by bottom wall shear stress (τwall=).For heat flux, the dispersive thermal flux 〈θ＇＇w＇＇〉 is normalized by bottom-wall heat flux (qwall= ?θτuτ).If one ne- glects the effects of viscous shear and molecular diffusion, the total value of fluxes (including the dispersive term and turbulent term) changes linearly from the bottom wall to the upper surface (mid- dle of the channel), which is shown with a solid narrow black line in Fig.6 .In case ”L-20” (almost neutral), because of roughness het- erogeneity and resulting secondary rolls, almost half of momentum flux is due to the dispersive term in the lower half of the domain (z/h＜0.5).In the cases L-4 and L-2, the convective stratification suppresses the share of the dispersive term to almost zero as the secondary rolls disappear.As the convective stratification is more intensified (the case ”L-1”), new secondary rolls, induced by the buoyancy effect, appear with opposite rotating direction.Therefore, the dispersive term again has a considerable share in both momen- tum and heat flux but it is more distributed in the whole domain.

Fig. 6. (a) Normalized mean spanwise-averaged wall-normal momentum fluxes.(b) Normalized mean spanwise-averaged wall-normal heat fluxes.

In summary, we investigated the effect of thermally con- vective/unstable stratification on secondary motions induced by spanwise-heterogeneous roughness in high-Reynolds-number tur- bulent boundary layers.For this purpose, the strength of convec- tive thermal stratification was systematically increased from MO lengthL/h= ?20 , which is almost equal to a neutral case, to MO lengthL/h= ?1 , which is an unstably/convective stratified case.Through this transition from the neutral stratified case to the con- vective stratified case, the results showed that the convective strat- ification (i.e.buoyancy effect) acts against the roughness-induced secondary rolls and HMPs/LMPs.As the magnitude of MO length decreases, the secondary rolls and HMPs/LMPs are weakened, and at MO lengthL/h= ?2 , they almost vanish.As the convective stratification is further intensified (MO lengthL/h= ?1), new sec- ondary rolls are appeared rotating in opposite direction; hence, the downward motion of flow over the high-roughness strip is re- placed by an upward motion.

An analysis of the streamwise mean vortex transport equation revealed that at MO lengthL/h= ?20 (almost neutrally stratified), the vortex is mainly generated due to the source term associated with the spatial gradient of Reynolds stresses.As the convective stratification is intensified, the source term associated with the buoyancy effect gets stronger and cancels out the generation by the spatial gradient of Reynolds stresses.At MO lengthL/h= ?2 , the vortices are almost disappeared, and in the case of MO lengthL/h= ?1 , the buoyancy effect is dominant and the vortices with an opposite sign appear inside the domain.

Comparison of RMS of dispersive normal velocity showed that these new large scale rolls induced by convective stratification are stronger than secondary rolls in the neutrally stratified case.The dispersive terms of heat flux and shear stress showed that the secondary motions in neutral case are taking a considerable share of flux in the lower half of the domain but in convective strati- fied case the distribution of dispersive term is spread through the whole domain.

Declaration of Competing Interest

The authors declare that they have no known competing finan- cial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgment

P.F.thanks the Aarhus University Research Foundation (AUFF) for the financial support.M.A.acknowledges the financial support from the Aarhus University Centre for Digitalisation, Big Data and Data Analytics (DIGIT).