2.1. Direct numerical simulation

For the present DNS, the governing equation is the fully three-dimensional Navier–Stokes equation for unsteady compressible flows, i.e.

(2.1)\begin{equation} \frac{\partial \boldsymbol{Q}}{\partial t}+\frac{\partial \boldsymbol{F}_{j}}{\partial x_{j}}=\frac{\partial \boldsymbol{F}_{{j,\nu}}}{\partial x_{j}}, \end{equation}

where $t$ is time, $x_{j}$ is the Cartesian coordinate in the $j$ direction and $\boldsymbol {Q}$, $\boldsymbol {F}_{j}$ and $\boldsymbol {F}_{{j,\nu }}$ represent the vectors of the conserved variables, convective fluxes and viscous fluxes, respectively. These are defined as

(2.2a–c)\begin{equation} \boldsymbol{Q}= \begin{bmatrix} \rho \\ \rho u_{i} \\ \rho e \\ \end{bmatrix}, \quad \boldsymbol{F}_{j}= \begin{bmatrix} \rho u_{j}\\ \rho u_{i} u_{j}+p\delta_{ij}\\ \left( \rho e+p \right)u_{j}\\ \end{bmatrix}, \quad \boldsymbol{F}_{{j,\nu}}= \begin{bmatrix} 0\\ \sigma_{ij}\\ \sigma_{ij} u_{i}+q_{j}\\ \end{bmatrix}, \end{equation}

with

(2.3a–c)\begin{equation} e=\frac{p}{\left( \gamma-1 \right)\rho}+\frac{u_{i}^{2}}{2}, \quad \sigma_{ij}=\mu\left( \frac{\partial u_{j}}{\partial x_{i}}+\frac{\partial u_{i}}{\partial x_{j}} \right)-\frac{2}{3}\mu \delta_{ij} \boldsymbol{\nabla} \boldsymbol{\cdot} \boldsymbol{u}, \quad q_{j}=\mu\frac{c_{p}}{\Pr}\frac{\partial T}{\partial x_{j}}. \end{equation}

Here, $\delta _{ij}$ is the Kronecker delta, $\boldsymbol {u} = [ u,v,w ]^{T}$ is the velocity vector, $p$ is the pressure, $\rho$ is the density and $c_{p}$ is the specific heat. The viscosity $\mu$ is calculated using Sutherland's law. The governing equations are closed by the state equation of a perfect gas. The specific heat ratio $\gamma$, gas constant $R$ and Prandtl number $Pr$ are set to 1.4, 287 and 0.72, respectively. The velocities are non-dimensionalized by the streamwise free-stream velocity $u_{\infty }$, and the other variables are non-dimensionalized by the corresponding free-stream values. The reference length is $L_{ref}=0.001$ m.

The governing equation is solved on a body-fitted grid using the finite-difference DNS code OpenCFD developed by Li, Fu & Ma (Reference Li, Fu and Ma2010). The third-order Runge–Kutta method (Jiang & Shu Reference Jiang and Shu1996) is used for time advancement. For spatial discretization, the convective flux vectors are computed by a seventh-order WENO-SYMBO scheme (Martín et al. Reference Martín, Taylor, Wu and Weirs2006) with Steger–Warming splitting (Steger & Warming Reference Steger and Warming1981). The viscous flux vectors are computed using the eight-order central difference method.

2.2. BiGlobal linear stability analysis

The flow variables $\boldsymbol {q}=[\rho, u, v, w, T]^{T}$ can be split into the mean flow $\bar {\boldsymbol {q}}$ and the small disturbance $\boldsymbol {q}'$, i.e.

(2.4)\begin{equation} \boldsymbol{q}(x,y,z,t) =\bar{\boldsymbol{q}}(x,y,z) +\boldsymbol{q}'(x,y,z,t). \end{equation}

By substituting equation (2.4) into the Navier–Stokes equation, subtracting the mean flow and ignoring the nonlinear terms, we obtain the governing equation of the small disturbance, namely the linearized Navier–Stokes (LNS) equation:

(2.5)\begin{align} &\boldsymbol{\varGamma}\frac{\partial \boldsymbol{q}'}{\partial t} +\boldsymbol{A}\frac{\partial \boldsymbol{q}'}{\partial x} +\boldsymbol{B}\frac{\partial \boldsymbol{q}'}{\partial y} +\boldsymbol{C}\frac{\partial \boldsymbol{q}'}{\partial z} +\boldsymbol{D}\boldsymbol{q}'\nonumber\\ &\quad =\boldsymbol{H}_{xx}\frac{\partial^{2} \boldsymbol{q'}}{\partial x^{2}}+\boldsymbol{H}_{yy}\frac{\partial^{2} \boldsymbol{q'}}{\partial y^{2}}+\boldsymbol{H}_{zz}\frac{\partial^{2} \boldsymbol{q'}}{\partial z^{2}}+\boldsymbol{H}_{xy}\frac{\partial^{2} \boldsymbol{q'}}{\partial x \partial y}+\boldsymbol{H}_{xz}\frac{\partial^{2} \boldsymbol{q'}}{\partial x \partial z}+\boldsymbol{H}_{yz}\frac{\partial^{2} \boldsymbol{q'}}{\partial y \partial z}, \end{align}

where $\boldsymbol {\varGamma }$, $\boldsymbol {A}$, $\boldsymbol {B}$, $\boldsymbol {C}$, $\boldsymbol {D}$, $\boldsymbol {H}_{xx}$, $\boldsymbol {H}_{yy}$, $\boldsymbol {H}_{zz}$, $\boldsymbol {H}_{xy}$, $\boldsymbol {H}_{xz}$ and $\boldsymbol {H}_{yz}$ are matrices that contain only the base flow variables. The non-zero elements in these matrices are listed in Appendix A.

For a steady flow that changes sharply in the wall-normal and spanwise directions but slowly in the streamwise direction, we obtain its local stability characteristics by introducing the quasi-parallel flow assumption, i.e. the mean flow variation in $x$ is negligible. Therefore, the small disturbance part can be expressed as

(2.6)\begin{equation} \boldsymbol{q}'(x,y,z,t)=\hat{\boldsymbol{q}}(y,z) \,{\rm{e}}^{{{\rm i}} (\alpha x-\omega t)}+{\rm{c.c.}}, \end{equation}

where $\hat {\boldsymbol {q}}$ is the shape function, $\alpha =\alpha _{r}+{\rm {i}}\alpha _{i}$ is the complex streamwise wavenumber, $\omega$ is the angular frequency and ${\rm {c.c.}}$ is the complex conjugate. The real part $\alpha _{r}$ is the streamwise wavenumber and the imaginary part $\alpha _{i}$ represents the disturbance's spatial growth rate. The corresponding streamwise wavelength and physical frequency are defined as

(2.7a,b)\begin{equation} \lambda_{x}=\frac{2 {\rm \pi}}{\alpha_{r}}, \quad f=\frac{\omega}{2 {\rm \pi}}. \end{equation}

Substituting equation (2.6) into (2.5), the LNS equation reduces to a polynomial eigenvalue problem:

(2.8)\begin{equation} (\mathcal{L}_{0}+\alpha \mathcal{L}_{1}+\alpha^{2} \mathcal{L}_{2} )\hat{\boldsymbol{q}}=0, \end{equation}

with

(2.9)\begin{equation} \left.\begin{array}{@{}c@{}} \displaystyle \mathcal{L}_{0}=\boldsymbol{D}-{\rm{i}} \omega \boldsymbol{\varGamma}+\boldsymbol{B}\dfrac{\partial}{\partial y}+\boldsymbol{C}\dfrac{\partial}{\partial z}-\boldsymbol{H}_{yy}\dfrac{\partial^{2}}{\partial y^{2}}-\boldsymbol{H}_{zz}\dfrac{\partial^{2}}{\partial z^{2}}-\boldsymbol{H}_{yz}\dfrac{\partial^{2}}{\partial y \partial z},\\ \displaystyle \mathcal{L}_{1}={\rm{i}}\left( \boldsymbol{A}-\boldsymbol{H}_{xy}\dfrac{\partial}{\partial y}-\boldsymbol{H}_{xz}\dfrac{\partial}{\partial z} \right),\\ \mathcal{L}_{1}=\boldsymbol{H}_{xx}. \end{array}\right\} \end{equation}

The above equation is discretized using the fourth-order central difference method in the $y$ and $z$ directions. At the wall, the Dirichlet condition is applied:

(2.10)\begin{equation} \hat{u}=\hat{v}=\hat{w}=\hat{T}=0, \quad y=0, \end{equation}

and $\hat {\rho }$ is solved for directly. A non-reflecting condition (Chang Reference Chang2004) is used in the far field, and a periodic condition is adopted for the two lateral boundaries. By solving this eigenvalue problem at a given frequency $\omega$ using the two-level orthogonal Arnoldi algorithm (Zhu Reference Zhu2005), we obtain the complex streamwise wavenumber $\alpha$ and the shape function $\hat {\boldsymbol {q}}$.

4.1. General flow features

The typical flow features in roughness-induced transitions were investigated through case $S_{1.0}$. Figures 3(a) and 3(b) plot the instantaneous $u$ at $y=0.5k_{r}$ and the time-averaged $C_{f}$ at the wall, respectively. From the velocity contours, we can find two evident high-speed streaks past the roughness element. The streaks first remain stable until they begin to break down at $x-x_{r} \approx 22 \delta _{r}$. Then, high-speed patterns appear in the streaks as well as in the low-speed region between the streaks. As the flow develops further downstream, the high-speed region expands in the spanwise direction, and a clear turbulent wedge appears. From the averaged $C_{f}$ contours, the roughness element first induces a high-wall-shear region. In the roughness wake region, the $C_{f}$ distribution exhibits a behaviour similar to that of the velocity distribution. The wall shear in the two streaks is greater. As the flow develops to the breakdown position, $C_{f}$ sharply increases and gradually expands in the spanwise direction.

Figure 3. (a) Instantaneous streamwise velocity at $y=0.5 k_{r}$ and (b) time-averaged skin friction coefficient $C_{f}$ at the wall for case $S_{1.0}$.

According to previous studies (De Tullio et al. Reference De Tullio, Paredes, Sandham and Theofilis2013; Lu et al. Reference Lu, Zeng, Liu, Liu and Yan2021), the roughness-induced transition at a supersonic Mach number is mainly induced by the instabilities in the wake region. Figure 4 shows different cross-stream slices in the wake region of case $S_{1.0}$. Here, the contours of the $Q$ criterion are plotted, which is defined as

(4.1)\begin{equation} Q={-}\frac{1}{2}\left[\left(\frac{\partial u}{\partial x}\right)^{2}+\left(\frac{\partial v}{\partial y}\right)^{2}+\left(\frac{\partial w}{\partial z}\right)^{2}\right]-\frac{\partial u}{\partial y}\frac{\partial v}{\partial x}-\frac{\partial u}{\partial z}\frac{\partial w}{\partial x}-\frac{\partial v}{\partial z}\frac{\partial w}{\partial y}, \end{equation}

with the reference value $Q_{ref}=u_{\infty }^{2}/L_{ref}^{2}$. At $x-x_{r}=6\delta _{r}$, we observe two pairs of vortices. The pair on the outside is the horseshoe vortex that is formed by the side of the roughness. The pair closest to the $z=0$ plane is the counter-rotating vortex created by the separated shear layer above the roughness. The $Q$ criterion states that the counter-rotating vortex is much stronger than the horseshoe vortex. The isolines of the streamwise velocity indicate that the boundary layer is strongly distorted by the counter-rotating vortex and is concentrated on the top region of the roughness element. The isolines of local shear, which is defined as

(4.2)\begin{equation} u_{s}=\sqrt{\left(\frac{\partial u}{\partial y}\right)^{2}+\left(\frac{\partial u}{\partial z}\right)^{2}}, \end{equation}

show that the distorted boundary layer generates a strong shear layer above the roughness element, and slight velocity fluctuations are observed in this region. As the flow develops downstream, the strength of the vortices decreases, and the boundary layer distortion also decreases; the separated shear layer remains strong and is gradually bent by the counter-rotating vortex. Moreover, the velocity fluctuations become much stronger, and the resulting distribution is highly consistent with that of the high-shear region. This phenomenon indicates that the interaction between the separated shear layer and counter-rotating vortex generates a high-shear region above the roughness element, thus leading to an increase in disturbance amplitude.

Figure 4. Slices of case $S_{1.0}$ at (a,b) $x-x_{r}=6\delta _{r}$, (c,d) $x-x_{r}=10\delta _{r}$ and (e,f) $x-x_{r}=14\delta _{r}$ coloured by (a,c,e) $Q$ criterion and (b,d,f) streamwise velocity fluctuation. The grey solid curves in (a,c,e) and (b,d,f) are isolines of streamwise velocity and local shear value, respectively. The black dashed curve is the outline of the roughness element.

To further investigate the instability in the roughness wake region, spectral analysis was used. As shown in figure 4, six different probes are adopted in the flow field to record the flow fluctuation, where probes 1, 3 and 5 are placed in the core of the counter-rotating vortex and probes 2, 4 and 6 are placed in the high-shear region at the $z=0$ plane. Figure 5 shows the spectral analysis results of the streamwise velocity fluctuations at these probes. Here, the dimensional physical frequency is non-dimensionalized to the Strouhal number St based on $u_{\infty }$ and $L_{ref}$. The six probes clearly confirmed the presence of a dominant mode with $St=0.462$ ($f=271$ kHz). This phenomenon further indicates that the dominant instability in the roughness wake results from the combination of the counter-rotating vortex and the separation shear layer.

Figure 5. The spectral analysis results of streamwise velocity fluctuation at (a) probes 1, 3 and 5 and (b) probes 2, 4 and 6 for case $S_{1.0}$.

BiGlobal linear stability analysis was then applied to case $S_{1.0}$. In this study, the computational grid for BiGlobal analysis is the same as that used for DNS. Figure 6 shows the instability characteristic at $x-x_{r}=10\delta _{r}$ with the dominant frequency $St=0.462$. Figure 6(a) plots a part of the eigenvalue spectrum. We can observe two evident unstable modes (marked by red and blue), and the real part of the shape function of the streamwise velocity ($\hat {u}_{r}$) is shown for the two modes in figures 6(b) and 6(c), respectively. For the mode in figure 6(b), the growth rate $-\alpha _{i}$ is large, and its shape function $\hat {u}_{r}$ shows a symmetric pattern with respect to the $z=0$ plane. For the mode in figure 6(c), the growth rate $-\alpha _{i}$ is low, and its shape function $\hat {u}_{r}$ shows an antisymmetric pattern with respect to the $z=0$ plane. In addition, the distributions of the two modes are located above the roughness element, corresponding to the distributions of velocity fluctuations and the high-shear region in figure 4(d).

Figure 6. BiGlobal analysis results at $x-x_{r}=10\delta _{r}$ with $St=0.462$ for case $S_{1.0}$: (a) eigenvalue spectrum and the real part of the shape function $\hat {u}$ for the (b) symmetric mode and (c) antisymmetric mode. The grey solid curves are isolines of the streamwise velocity. The black dashed curve is the outline of the roughness element.

Figure 7 further plots the growth rate and amplification factor $N$ at different streamwise positions and frequencies. Here, the $N$ factor is calculated integrally in the streamwise direction for each frequency $St$ as (Groskopf & Kloker Reference Groskopf and Kloker2016)

(4.3)\begin{equation} N(St,x)=\int_{x_{0}}^{x}-\alpha_{i}(x)\, {\rm{d}} x, \end{equation}

where $x_{0}$ is the start position for the integration. This operation is similar to that of the traditional linear stability theory and has also been used in BiGlobal analysis of roughness-induced transitions (Groskopf & Kloker Reference Groskopf and Kloker2016). For the symmetric mode, its maximum growth rate is approximately $-\alpha _{i}L_{ref}\approx 1.1$ and is located in the region $5.5\delta _{r}< x-x_{r}<7.5\delta _{r}$ with $0.49< St<0.58$. For the antisymmetric mode, the maximum growth rate is only approximately 0.3. In addition, the antisymmetric mode becomes stable at $x-x_{r}>18\delta _{r}$. This phenomenon indicates that the symmetric mode has a higher growth rate and a wider unstable region than the antisymmetric mode. The maximum amplification factor of the symmetric mode exceeds 6 at $x-x_{r}>21\delta _{r}$ within $0.41< St<0.51$. However, because of the much lower growth rate, the $N$ factor of the antisymmetric mode is lower than that of the symmetric mode at the same streamwise location and frequency. Because the antisymmetric mode is almost stable at $x-x_{r}>16\delta _{r}$, the amplification factor in this region has almost no growth. The above analysis further confirms that the symmetric mode is much stronger than the antisymmetric mode and is therefore responsible for the final transition to turbulence.

Figure 7. (a,b) Growth rate and (c,d) amplification factor of the (a,c) symmetric mode and (b,d) antisymmetric mode at different frequencies and streamwise positions for case $S_{1.0}$.

Through the amplification factor, we can determine the most unstable mode, which has the maximum $N$ factor. Figure 8 shows the growth of the streamwise velocity fluctuation from DNS and compares it with the most unstable symmetric ($St=0.477$, $f=280$ kHz) and antisymmetric ($St=0.366$, $f=215$ kHz) modes. The amplitude of the symmetric mode is clearly much larger than that of the antisymmetric mode and shows much better agreement with the DNS. In addition, the most unstable frequency identified by BiGlobal analysis is very close to that identified by the spectral analysis. Overall, the above results indicate that BiGlobal linear stability analysis can simulate the evolution of the instability in the wake region with reasonable accuracy.

Figure 8. Comparison of streamwise velocity fluctuation amplitudes between DNS and symmetric and antisymmetric modes from BiGlobal analysis for case $S_{1.0}$.

4.2. Wall temperature effects

The wall temperature effect is studied through comparisons between cases $S_{0.9}$, $S_{1.1}$ and $S_{1.2}$. Figure 9(a–d) first shows the general instantaneous flow features for the three cases. For examining if the transition occurs in case $S_{1.2}$, the computation is performed on a longer well-resolved domain with $L_{x}=310 \delta _{r}$ and $n_{x}=3240$, and the sizes and grid numbers in wall-normal and spanwise directions are not changed. Note that figure 9(d) is the spatial continuation of figure 9(c).

Figure 9. Instantaneous streamwise velocity at $y=0.5k_{r}$ for (a) case $S_{0.9}$, (b) case $S_{1.1}$ and (c,d) case $S_{1.2}$ and time- and spanwise-averaged (e) skin friction coefficient and ( f) momentum thickness Reynolds number.

Clearly, compared to case $S_{1.0}$, a lower wall temperature results in an earlier transition to turbulence, while a higher wall temperature significantly delays the transition. For case $S_{0.9}$, the streaks become stronger and quickly break down at $x-x_{r} \approx 15 \delta _{r}$. For case $S_{1.1}$, the strength of the streaks decreases, and the breakdown position moves downstream to $x-x_{r}\approx 45\delta _{r}$. When $T_{w}$ increases to $1.2 T_{aw}$ in case $S_{1.2}$, the high-speed streaks further weaken and remain steady in the short computational domain, and the breakdown finally occurs at $x-x_{r}\approx 190\delta _{r}$. Figure 9(e,f) further plots the time- and spanwise-averaged $C_{f}$ and momentum thickness Reynolds number $Re_{\theta }$ for these cases. It is clear that as the wall temperature increases, the transition onset position is delayed significantly. Taking case $S_{1.0}$ as the baseline, the transition onset in case $S_{0.9}$ moves upstream by approximately $7\delta _{r}$ and that in case $S_{1.1}$ moves downstream by approximately $15\delta _{r}$. In case $S_{1.2}$, we scale the horizontal axis to display the complete distribution of $C_{f}$ (see the green label). Obviously, $C_{f}$ exhibits laminar behaviour in a long distance until it begins to increase sharply at $x-x_{r}\approx 190\delta _{r}$. A similar phenomenon is shown by the distribution of $Re_{\theta }$. However, the rapid increase in $Re_{\theta }$ occurs slightly later than that in $C_{f}$, indicating that the growth of the boundary layer lags the increase in skin friction.

The reason why an increasing wall temperature can suppress the transition is discussed below. Figure 10 shows the streamwise velocity on the $z=0$ plane in the upstream region of the roughness. Obviously, with a higher wall temperature, the inflow boundary layer is lifted. Using the flow variables computed by the similarity equations (White Reference White2006), we can compute the roughness Reynolds number $Re_{k}=\rho _{k} U_{k} k_{r}/\mu _{k}$. Here, the subscript $k$ represents the quantity measured at $y=k_{r}$ in a smooth laminar boundary layer. For cases $S_{0.9}$, $S_{1.0}$, $S_{1.1}$ and $S_{1.2}$, $Re_{k}$ are 522, 420, 342 and 282, respectively. This phenomenon shows that a heating wall can significantly reduce the roughness Reynolds number and suppress the transition. In addition, the streamlines in figure 10(b,d,f) show that a heating wall weakens the vortex system upstream of the roughness, and the separation region also decreases.

Figure 10. Streamwsie velocity and streamlines on the $z=0$ plane: (a,b) case $S_{0.9}$, (c,d) case $S_{1.0}$ and (e,f) case $S_{1.1}$.

The changes in the roughness wake region are shown in figure 11. A comparison between cases $S_{0.9}$ and $S_{1.1}$ shows that a higher wall temperature significantly weakens the strength of the horseshoe and counter-rotating vortex pairs in the wake region, reducing the boundary layer distortion. Thus, the local shear strength decreases substantially, indicating that the strength of the separated shear layer decreases. As discussed in § 4.1, the transition induced by the roughness element is mainly caused by the interaction between the vortex system and separated shear layer; therefore, a higher wall temperature may suppress the instabilities in the roughness region and delay the transition.

Figure 11. Slices of (a,c) case $S_{0.9}$ and (b,d) case $S_{1.1}$ at (a,b) $x-x_{r}=6\delta _{r}$ and (c,d) $x-x_{r}=10\delta _{r}$ coloured by the $Q$ criterion. The grey solid curves on the left half are isolines of the streamwise velocity and the black solid curves on the right half are isolines of the local shear value. The black dashed curve is the outline of the roughness element.

Using BiGlobal linear stability analysis, the neutral curves of the symmetric and antisymmetric modes at $x-x_{r}=10\delta _{r}$ are plotted in figure 12(a). Obviously, compared with the baseline case $S_{1.0}$, a lower wall temperature in case $S_{0.9}$ results in a higher growth rate and wider unstable region for the symmetric and antisymmetric modes. However, a higher wall temperature significantly suppresses the unstable modes. For cases $S_{1.1}$ and $S_{1.2}$, for the symmetric mode, the growth rate decreases, and the unstable region is narrowed. The antisymmetric mode is fully suppressed and becomes fully stable. In addition, as $T_{w}$ increases, the neutral curve moves towards the low-frequency region. The frequency of the disturbance with the largest growth rate also decreases significantly. The distributions of $\hat {u}_{r}$ are illustrated in figure 12(b–e) for the disturbance with the largest growth rate at $x-x_{r}=10\delta _{r}$. Although the wall temperature changes, the shape of the unstable modes changes little compared with that in the baseline case $S_{1.0}$. The peak disturbance is still located in the separated shear layer region above the roughness element. This phenomenon indicates that varying the wall temperature does not change the type of dominant instability but rather only influences the growth rate and the unstable region.

Figure 12. BiGlobal results for cases with different wall temperatures at $x-x_{r}=10\delta _{r}$. (a) Neutral curves and (b–e) the real part of the shape function $\hat {u}$: (b) symmetric mode at $St=0.680$ in case $S_{0.9}$, (c) antisymmetric mode at $St=0.680$ in case $S_{0.9}$, (d) symmetric mode at $St=0.407$ in case $S_{1.1}$ and (e) symmetric mode at $St=0.264$ in case $S_{1.2}$. The grey solid curves are isolines of the streamwise velocity. The black dashed curve is the outline of the roughness element.

5.1. Upstream control

Having obtained an understanding of the flow features of the roughness-induced transition and the main instability characteristics in the roughness wake region, as well as the wall temperature effect on the transition, we now focus on transition control through upstream and downstream wall heating strips. Figure 13(a–e) depicts the instantaneous streamwise velocity distributions for cases with upstream control effects. We find that with the upstream wall heating effect, the high-speed streaks in the wake region remain stable over a longer distance compared with those in case $S_{1.0}$. In case $S_{-6}L_{10}T_{2.0}$, with a control strip that has a width of $L_{strip}=10k_{r}$ and a heating temperature of $T_{strip}=2.0T_{aw}$, the breakdown of the wake moves downstream to $x-x_{r} \approx 30 \delta _{r}$. In case $S_{-6}L_{10}T_{2.5}$, as the temperature of the heating strip increases to $T_{strip}=2.5T_{aw}$, the transition is further delayed to $x-x_{r} \approx 34 \delta _{r}$. In case $S_{-6}L_{20}T_{2.0}$, the width of the strip increases to $L_{strip}=20\delta _{r}$, and the transition moves further downstream to $x-x_{r} \approx 36\delta _{r}$ compared with that in case $S_{-6}L_{10}T_{2.0}$. In addition, the effect of the location of the control strip was also studied. In cases $S_{-10}L_{10}T_{2.0}$ and $S_{-10}L_{20}T_{2.0}$, although the control strip has been moved upstream to $x_{strip}-x_{r}=-10\delta _{r}$, the transition scenario shows only slight differences compared with cases $S_{-6}L_{10}T_{2.0}$ and $S_{-6}L_{20}T_{2.0}$, respectively. The above comparison shows that the transition suppression is enhanced with a wider control strip or a higher control temperature, and the location of the control strip weakly influences the transition control effect. Figures 13( f) and 13(g) plot the time- and spanwise-averaged $C_{f}$ and $Re_{\theta }$ for the cases with upstream control strips, respectively. Generally, the comparison of $C_{f}$ shows that the upstream wall heating strip can suppress the transition, but the suppression effect is not very significant, and the flow develops to full turbulence in all the cases. The suppression effect is enhanced with increasing heating temperature and width of the control strip. The location of the upstream control strip has little effect on the transition onset position. The evolution of $Re_{\theta }$ has a similar tendency to that of $C_{f}$.

Figure 13. Instantaneous streamwisse velocity at $y=0.5 k_{r}$ for (a) case $S_{-6}L_{10}T_{2.0}$, (b) case $S_{-6}L_{10}T_{2.5}$, (c) case $S_{-6}L_{20}T_{2.0}$, (d) case $S_{-10}L_{10}T_{2.0}$ and (e) case $S_{-10}L_{20}T_{2.0}$ and time- and spanwise-averaged ( f) skin friction coefficient and (g) momentum thickness Reynolds number.

The flow fields in the wake region are plotted in figure 14. Compared with the baseline case $S_{1.0}$, the main flow structure of the wake changes slightly with the upstream wall heating strip in case $S_{-6}L_{10}T_{2.0}$. However, the vortex system and separation shear layer become weaker. The distortion of the flow profile and the velocity fluctuations are also diminished. As the heating temperature or width of the control strip increases, the strength of the wake structure further decreases. Therefore, the interaction between the separation shear layer and counter-rotating vortex is suppressed, and the instabilities in the wake region are suppressed as well. As the transition is induced by the instabilities in the wake region, the transition is delayed by the upstream wall heating strip.

Figure 14. Slices of (a,b) case $S_{-6}L_{10}T_{2.0}$, (c,d) case $S_{-6}L_{10}T_{2.5}$ and (e,f) case $S_{-6}L_{20}T_{2.0}$ at $x-x_{r}=10\delta _{r}$ coloured by (a,c,e) $Q$ criterion and (b,d,f) streamwise velocity fluctuation. The grey solid curves in (a,c,e) and (b,d,f) are isolines of streamwise velocity and local shear value, respectively. The black dashed curve is the outline of the roughness element.

Compared with the cases with a whole heating wall, the upstream wall heating strip does not directly influence the flow structures in the roughness wake region but rather influences the inflow boundary layers. Figure 15 plots the streamwise velocity distributions in the $z=0$ plane. The upstream wall heating strip can directly lift up the inflow, thereby decreasing $Re_{k}$ and weakening the flow structure in the wake region. In addition, upstream wall heating strips can weaken the flow separation in front of the roughness element. Using DNS of the upstream wall heating strip but without the roughness element, the flow parameters at $x=x_{r}$ and $y=k_{r}$ are listed in table 4. These parameters reveal that the local Mach number, density and streamwise velocity decrease with increasing temperature. This variation results in a lower $Re_{k}$. In addition, $Re_{k}$ in cases $S_{-6}L_{10}T_{2.0}$ and $S_{-10}L_{10}T_{2.0}$ only have a slight difference, and the difference between cases $S_{-6}L_{20}T_{2.0}$ and $S_{-10}L_{20}T_{2.0}$ is a little more obvious. However, these data are extruded from undisturbed laminar boundary layer profiles; in the actual transition cases, the roughness element will hinder the development of the flow and therefore the situation will be a little different.

Figure 15. Streamwise velocity and streamlines in the $z=0$ plane: (a,b) case $S_{1.0}$, (c,d) case $S_{-6}L_{10}T_{2.0}$ and (e,f) case $S_{-6}L_{20}T_{2.0}$.

Table 4. Parameters of upstream control cases at $x=x_{r}$ and $y=k_{r}$ in an undisturbed laminar boundary layer.

To further investigate the effect of upstream wall heating on the instability in the roughness wake region, BiGlobal linear stability analysis was performed. Figure 16 shows part of the eigenvalue spectrum for cases $S_{-6}L_{10}T_{2.0}$, $S_{-6}L_{10}T_{2.5}$ and $S_{-6}L_{20}T_{2.0}$ at $x-x_{r}=10\delta _{r}$ with $St=0.462$. In case $S_{-6}L_{10}T_{2.0}$, we can still observe two obvious unstable modes, namely the symmetric (marked in blue) and antisymmetric (marked in red) modes. However, compared with the results of case $S_{1.0}$ in figure 6(a), the growth rate $-\alpha _{i}$ significantly decreases and the streamwise wavenumber $\alpha _{r}$ increases slightly. In case $S_{-6}L_{10}T_{2.5}$, as the heating temperature increases, the growth rate of the symmetric mode further decreases, and the antisymmetric mode vanishes. In case $S_{-6}L_{20}T_{2.0}$, the width of the strip increases, the symmetric mode is also suppressed and the antisymmetric mode vanishes. This phenomenon indicates that the upstream wall heating strip can effectively suppress both the symmetric and antisymmetric modes.

Figure 16. Eigenvalue spectrum at $x-x_{r}=10\delta _{r}$ with $St=0.462$ for (a) case $S_{-6}L_{10}T_{2.0}$, (b) case $S_{-6}L_{10}T_{2.5}$ and (c) case $S_{-6}L_{20}T_{2.0}$.

The results for the growth rate and amplification factor for the symmetric mode are presented in figure 17. Compared with the results for case $S_{1.0}$ in figure 7(a), the maximum growth rate in case $S_{-6}L_{10}T_{2.0}$ is significantly lower, but the unstable region becomes larger. In case $S_{-6}L_{10}T_{2.5}$, as $T_{strip}$ increases to 2.5 $T_{aw}$, the maximum growth rate is further suppressed, but the symmetric mode maintains a growth rate of $-\alpha _{i}L_{ref}\approx 0.4$ for a long distance. In case $S_{-6}L_{20}T_{2.0}$, $L_{strip}$ increases to $20k_{r}$; however, although the maximum growth rate further decreases, the symmetric mode still has a clearly unstable growth rate over a long distance. From the distributions of the amplification factor, its maximum value for the cases with upstream control strips achieves a similar level of approximately 5–5.5 before the final breakdown to turbulence. In addition, the unstable mode with the largest growth rate or amplification factor is shifted to the lower-frequency region compared with the case $S_{1.0}$.

Figure 17. (a–c) Growth rate and (d–f) amplification factor for the symmeyric mode of (a,d) case $S_{-6}L_{10}T_{2.0}$, (b,e) case $S_{-6}L_{10}T_{2.5}$ and (c,f) case $S_{-6}L_{20}T_{2.0}$ at different frequencies and streamwise positions.

As discussed by De Tullio et al. (Reference De Tullio, Paredes, Sandham and Theofilis2013), the growth of the symmetric mode is mainly due to the wall-normal and spanwise components of the local shear layer. To analyse this effect, the contributions of the wall-normal and spanwise shear components to the production of the disturbance kinetic energy in the cases with upstream wall heating strips are analysed. The production term is computed as

(5.1)\begin{equation} P=\overbrace{-\bar{\rho}{\rm Re}(\hat{u}\hat{v}^{{{\dagger}}})\frac{\partial \bar{u}}{\partial y}}^{P_{y}} \overbrace{-\bar{\rho}{\rm Re}(\hat{u}\hat{w}^{{{\dagger}}})\frac{\partial \bar{u}}{\partial z}}^{P_{z}}. \end{equation}

Here, ${\dagger}$ represents the complex conjugate, ${\rm Re}$ denotes the real part and $P_{y}$ and $P_{z}$ indicate the production terms caused by the wall-normal and spanwise shear, respectively. The distributions of $P_{y}$ for the most amplified modes are shown in figure 18, with the reference value defined as $P_{ref}=\rho _{\infty }u_{\infty }|\hat {u}|_{max}^{2}/L_{ref}$. Here, the most amplified mode is selected by the mode that has the largest $N$ factor in figure 17(d–f). These modes have frequencies $St=0.392$, 0.359 and 0.342 for cases $S_{-6}L_{10}T_{2.0}$, $S_{-6}L_{10}T_{2.5}$ and $S_{-6}L_{20}T_{2.0}$, respectively. At the three different streamwise locations, $P_{y}$ always has a positive effect, although the magnitude of $P_{y}$ gradually decreases downstream. This indicates that the wall-normal shear always contributes to the growth of the symmetric mode. In addition, as the suppression effect caused by the wall heating strip increases, the peak value of $P_{y}$ decreases significantly. This means that the wall heating strip can weaken the contributions of the wall-normal shear to the growth of the symmetric mode.

Figure 18. Production term $P_{y}$ of disturbance kinetic energy at (a,d,g) $x-x_{r}=3\delta _{r}$, (b,e,h) $x-x_{r}=6\delta _{r}$ and (c,f,i) $x-x_{r}=9\delta _{r}$ for (a–c) case $S_{-6}L_{10}T_{2.0}$, (d–f) case $S_{-6}L_{10}T_{2.5}$ and (g–i) case $S_{-6}L_{20}T_{2.0}$. The grey solid curves are isolines of the streamwise velocity. The black dashed curve is the outline of the roughness element.

Figure 19 further plots the distributions of $P_{z}$. Compared with $P_{y}$ at the same position, $P_{z}$ shows a different behaviour. That is, $P_{y}$ has clear negative peaks at the first two streamwise locations. This shows that spanwise shear retards the growth of the symmetric mode in the near-roughness region. However, different results are obtained for $x-x_{r}=9\delta _{r}$. Term $P_{z}$ has a positive distribution at $x-x_{r}=9\delta _{r}$ for case $S_{-6}L_{10}T_{2.0}$, although the peak value of $P_{z}$ is much smaller than that of $P_{y}$. This indicates that spanwise shear accelerates the growth of the symmetric modes in the downstream region. However, this positive contribution becomes weaker when the transition suppression increases in cases $S_{-6}L_{10}T_{2.5}$ and $S_{-6}L_{20}T_{2.0}$, as shown in figure 19( f,i). In particular, the positive peak is less intense than the negative peak in figure 19(i). The above results indicate that the upstream wall heating strip can also suppress the contribution of the spanwise shear to the growth of the symmetric mode.

Figure 19. Production term $P_{z}$ of disturbance kinetic energy at (a,d,g) $x-x_{r}=3\delta _{r}$, (b,e,h) $x-x_{r}=6\delta _{r}$ and (c,f,i) $x-x_{r}=9\delta _{r}$ for (a–c) case $S_{-6}L_{10}T_{2.0}$, (d–f) case $S_{-6}L_{10}T_{2.5}$ and (g–i) case $S_{-6}L_{20}T_{2.0}$. The grey solid curves are isolines of the streamwise velocity. The black dashed curve is the outline of the roughness element.

5.2. Downstream control

The downstream control effect is studied below. The instantaneous streamwise velocities at $y=0.5k_{r}$ are shown in figure 20(a–d). In case $S_{6}L_{10}T_{2.0}$, the high-speed streaks remain steady until $x-x_{r}> 36\delta _{r}$, which is later than that in the upstream control case for the same heating temperature and strip width. As the heating temperature $T_{strip}$ increased to $2.5T_{aw}$, the transition further decreased to $x-x_{r}\approx 42\delta _{r}$. The effect of the strip width is also studied. As shown by the DNS result, the wake structure in case $S_{6}L_{20}T_{2.0}$ remains stable at a longer distance than that in the other cases. This indicates that a wider strip has a much more significant transition-delaying effect. The strip location is studied through case $S_{10}L_{20}T_{2.0}$, in which the breakdown position of the streaks is much earlier than that in case $S_{6}L_{20}T_{2.0}$. This indicates that later wall heating in the downstream position may weaken the transition suppression effect. Generally, compared with the cases with an upstream control strip (figure 13), we find that the downstream control strip is much more effective at suppressing the transition, but the location of the strip should be close to the roughness element. Figures 20(e) and 20( f) plot the time- and spanwise-averaged $C_{f}$ and $Re_{\theta }$ distributions for cases with downstream control strips, respectively. Generally, downstream wall heating has little influence on the flow upstream of the roughness element. Downstream of the roughness element, the heating strip causes sharp increases in $C_{f}$ and $Re_{\theta }$. Behind the heating strip, the values of $C_{f}$ and $Re_{\theta }$ recover to the laminar level. Then, the sharp and continuous increases in $C_{f}$ and $Re_{\theta }$ indicate the final transition to turbulence. According to the distribution of $C_{f}$, cases $S_{6}L_{10}T_{2.0}$ and $S_{6}L_{10}T_{2.5}$ exhibit full turbulence, and the flow in case $S_{6}L_{20}T_{2.0}$ is still in the transitional region. In addition, there are almost no differences in the distributions of $C_{f}$ and $Re_{\theta }$ between cases $S_{1.0}$ and $S_{10}L_{20}T_{2.0}$.

Figure 20. Instantaneous streamwise velocity at $y=0.5 k_{r}$ for (a) case $S_{6}L_{10}T_{2.0}$, (b) case $S_{6}L_{10}T_{2.5}$, (c) case $S_{6}L_{20}T_{2.0}$ and (d) case $S_{10}L_{20}T_{2.0}$ and time- and spanwise-averaged (e) skin friction coefficient and ( f) momentum thickness Reynolds number.

As discussed in §§ 4.2 and 5.1, a whole heating wall or an upstream wall heating strip can weaken the flow structure in the roughness wake and therefore delay the transition. To study the influence of the downstream wall heating strip on the wake region, slices at $x-x_{r}=10\delta _{r}$ are presented in figure 21. Clearly, compared with the upstream control case with the same $T_{strip}$ and $L_{strip}$, the shape and strength of the wake vortex system in the downstream control case change slightly, but the counter-rotating vortex pair is pushed away from the $z=0$ plane, and the isolines of streamwise velocity show that the boundary layer distortion weakens. The local shear layer indicates that the strength of the separated shear layer decreases. Therefore, the interaction between the vortex and the separated shear layer weakens, resulting in lower streamwise velocity fluctuations. The comparison also shows that a higher $T_{strip}$ or a larger $L_{strip}$ in the downstream control strip can more effectively reduce the separation shear layer and suppress the development of the velocity fluctuation. However, when a wide strip moves downstream, the boundary layer distortion and vortex system become strong, therefore resulting in a high-strength shear layer and high-amplitude streamwise velocity fluctuations.

Figure 21. Slices of (a,b) case $S_{6}L_{10}T_{2.0}$, (c,d) case $S_{6}L_{10}T_{2.5}$, (e,f) case $S_{6}L_{20}T_{2.0}$ and (g,h) case $S_{10}L_{20}T_{2.0}$ at $x-x_{r}=10\delta _{r}$ coloured by (a,c,e,g) $Q$ criterion and (b,d,f,h) streamwise velocity fluctuation. The grey solid curves in (a,c,e,g) and (b,d,f,h) are isolines of the streamwise velocity and local shear value, respectively. The black dashed curve is the outline of the roughness element.

The discussion in § 5.1 also shows that the transition suppression effect in upstream control cases occurs through lifting the inflow boundary layer rather than by directly changing the wake structure. However, the downstream control strip has different mechanisms. Figure 22 shows the streamwise vorticity $\omega _{x}=\partial w/\partial y-\partial v/\partial z$ at $z=0.3\delta _{r}$, with the reference value defined as $\omega _{ref}=u_{\infty }/L_{ref}$. In case $S_{1.0}$, the streamwise vortex retains a relatively large strength in the wake region. In case $S_{6}L_{10}T_{2.0}$, the wall heating strip lifts up the streamwise vortex and pushes it away from the wall. This lift-up effect compresses the streamwise vortex in the wall-normal direction and weakens it considerably in the downstream region. As $T_{strip}$ increases to $2.5T_{aw}$, the lifting effect is enhanced, and the streamwise vortex is further weakened. For a wider control strip, this effect becomes much more obvious, and the streamwise vortex further becomes weak. However, when the wide strip moves downstream, the situation is different. Because the flow has developed at a longer distance, the boundary layer has grown thick enough, and the lifting up to the streamwise vortex becomes much weaker. Thus, although $\omega _{x}$ has a lower peak value, it remains moderate for a sufficiently long distance.

Figure 22. Contours of streamwise vorticity $\omega _{x}$ at $z=0.3\delta _{r}$ plane for (a) case $S_{1.0}$, (b) case $S_{6}L_{10}T_{2.0}$, (c) case $S_{6}L_{10}T_{2.5}$, (d) case $S_{6}L_{20}T_{2.0}$ and (e) case $S_{10}L_{20}T_{2.0}$.

The weakening of the streamwise vortex may originate from different mechanisms. For further investigation, we use vorticity transport analysis according to the following equation:

(5.2)\begin{equation} \frac{\partial}{\partial t}\left(\frac{1}{2}\omega_{i}^{2}\right)+\boldsymbol{u} \boldsymbol{\cdot}\boldsymbol{\nabla} \left(\frac{1}{2}\omega_{i}^{2}\right)=\omega_{i}\boldsymbol{\omega} \boldsymbol{\cdot} \boldsymbol{\nabla} u_{i} +\nu\left[ \nabla^{2} \left( \frac{1}{2} \omega_{i}^{2}\right)-|\boldsymbol{\nabla} \omega_{i}|^{2} \right]. \end{equation}

Here, $\boldsymbol {\omega }=[\omega _{x}, \omega _{y}, \omega _{z}]^\textrm {{T}}$ is the vorticity vector. The production term is $P_{i}=\omega _{i}\boldsymbol {\omega }\boldsymbol {\cdot } \boldsymbol {\nabla } u_{i}=\sum _{j} P_{ij}$, its component $P_{ii}=\omega _{i}^{2}\partial u_{i}/\partial x_{i}$ indicates the stretch of $\omega _{i}$ and $P_{ij}=\omega _{i}\omega _{j}\partial u_{i}/\partial x_{j}$ represents the transfer from $\omega _{j}$ to $\omega _{i}$. The dissipation term is $D_{i}=\nu |\boldsymbol {\nabla } \omega _{i}^{2}|$.

The distributions of the production term $P_{xx}$, $P_{yx}$, $P_{zx}$ and dissipation term $D_{x}$ are shown in figure 23 for cases $S_{1.0}$ and $S_{6}L_{10}T_{2.0}$. The results show that the wall heating strip causes an evident peak in streamwise stretch production. This leads to a temporary increase in $\omega _{x}$, as shown in figures 22(b). However, the distribution of $D_{x}$ indicates that the wall heating strip also causes a larger dissipation peak. We can also see that in case $S_{1.0}$, $P_{yx}$ or $P_{zx}$ do not show clear distributions. For $S_{6}L_{10}T_{2.0}$, significant peaks are observed over the wall heating strip region. The $P_{yx}$ and $P_{zx}$ peaks indicate that the transfer of $\omega _{x}$ to $\omega _{y}$ and $\omega _{z}$ is strong, significantly contributing to weakening the vortex system in the wake region.

Figure 23. Contours of the vorticity transport term (a,b) $P_{xx}$, (c,d) $D_{x}$, (e,f) $P_{yx}$ and (g,h) $P_{zx}$ at $z=0.3\delta _{r}$ for (a,c,e,g) case $S_{1.0}$ and (b,d,f,h) case $S_{6}L_{10}T_{2.0}$.

From the formulation of $D_{x}$, the local wall heating strip may increase $D_{x}$ by enhancing the viscosity or enhancing the gradient of $\omega _{x}$. To find which way is more important, we plot the kinematic viscosity $\nu$ and the derivatives of streamwise vorticity $\partial \omega _{x}/\partial x$, $\partial \omega _{x}/\partial y$ and $\partial \omega _{x}/\partial z$ for cases $S_{1.0}$ and $S_{6}L_{10}T_{2.0}$ in figure 24. The yellow circle symbol in each panel represents a probe where $D_{x}$ has the peak value. Comparing the viscosity in figures 24(a) and 24(b), we can see that the wall heating strip causes a large region of high viscosity, but at the location of the yellow probe, the viscosity does not have very obvious differences in the two cases. Next, we focus on the gradients of $\omega _{x}$. In figure 24(c,d), $\partial \omega _{x}/\partial x$ is near zero at the probe in case $S_{1.0}$, and significant peaks are observed over the wall heating strip region in case $S_{6}L_{10}T_{2.0}$. In figure 24(e,f), $\partial \omega _{x}/\partial y$ has a small value at the probe in case $S_{1.0}$, but a much larger peak value is observed over the wall heating strip region in case $S_{6}L_{10}T_{2.0}$, and this peak region is very similar to the distribution of $D_{x}$ in figure 23(b). In figure 24(g,h), $\partial \omega _{x}/\partial z$ has similar values at the probe in cases $S_{1.0}$ and $S_{6}L_{10}T_{2.0}$, and this value is much smaller than that of $\partial \omega _{x}/\partial y$. Overall, the above analysis indicates that the wall heating affects the transportation of $\omega _{x}$ mainly through causing gradients in the wall-normal direction.

Figure 24. Contours of the (a,b) kinematic viscosity and (c,d) streamwise, (e,f) wall-normal and (g,h) spanwise derivatives of the streamwise vorticity in (a,c,e,g) case $S_{1.0}$ and (b,d,f,h) case $S_{6}L_{10}T_{2.0}$.

BiGlobal analysis is conducted to further investigate the instability characteristics in the cases with downstream wall heating strips. A part of the eigenvalue spectrum at $St=0.462$ is plotted in figure 25. For case $S_{6}L_{10}T_{2.0}$, symmetric and antisymmetric modes are observed, and their growth rates are lower than those in case $S_{1.0}$. However, compared with the upstream control case $S_{6}L_{10}T_{2.0}$, the symmetric mode in case $S_{6}L_{10}T_{2.0}$ is further suppressed, but the growth rate of the antisymmetric mode is stronger. In addition, the streamwise wavenumbers also clearly decrease. When $T_{strip}$ increases to $2.5T_{aw}$, the symmetric and antisymmetric modes still exist in the downstream control case, while the antisymmetric mode vanishes in the corresponding upstream control case. As $L_{strip}$ increases to $20\delta _{r}$, the two modes are still unstable, although their growth rate is further reduced, and the antisymmetric mode approaches the stable region. In addition, the streamwise wavenumber decreases. Overall, we can summarize that the downstream strip suppresses the transition by mainly stabilizing the symmetric mode, and the antisymmetric mode is affected less strongly.

Figure 25. Eigenvalue spectra at $x-x_{r}=10\delta _{r}$ with $St=0.462$ for (a) case $S_{6}L_{10}T_{2.0}$, (b) case $S_{6}L_{10}T_{2.5}$ and (c) case $S_{6}L_{20}T_{2.0}$.

The results for the growth rate and amplification factor for the symmetric mode are shown in figure 26. Compared with those in the upstream control cases, the growth rates in the downstream control cases had larger peak values, but the unstable region narrowed significantly. In particular, for $S_{6}L_{10}T_{2.0}$, the maximum growth rate is greater than 1.2, which is nearly 1.5 times greater than that for $S_{-6}L_{10}T_{2.0}$. However, behind the peak region, the growth rate quickly decreases to a much lower value as the flow develops downstream. Therefore, the $N$ factor needs a long distance to increase to approximately 6, as shown in figure 26(d). In case $S_{6}L_{10}T_{2.5}$, the maximum growth rate remains at a value of 1.2, but the unstable region further narrows, and the growth rate behind the control strip decreases. Therefore, a longer distance is needed to induce the transition. This is also confirmed by the amplification factor distributions in figure 26(e), which shows that the $N$ factor increases to approximately 6 at $x-x_{r}\approx 40\delta _{r}$. In case $S_{6}L_{20}T_{2.0}$, a wider control strip has a much stronger stabilization effect. The maximum growth rate decreases to approximately 1.0, and the unstable region also becomes narrower. Therefore, the transition is delayed further downstream. However, the maximum amplification factor in this case reaches approximately 6.7 in front of the transition, which is slightly larger than the value of approximately 6.0 in cases $S_{6}L_{10}T_{2.0}$ and $S_{6}L_{10}T_{2.5}$. This difference may be due to the neglect of the non-parallel effect in the roughness wake region in the BiGlobal linear stability analysis.

Figure 26. (a–c) Growth rate and (d–f) amplification factor for the symmetric mode of (a,d) case $S_{6}L_{10}T_{2.0}$, (b,e) case $S_{6}L_{10}T_{2.5}$ and (c,f) case $S_{6}L_{20}T_{2.0}$ at different frequencies and streamwise positions.

The effects of wall-normal and spanwise shears on the disturbance production of symmetric modes in cases with downstream wall heating strips are analysed below. The results are shown at $St=0.407$, 0.375 and 0.359 for cases $S_{6}L_{10}T_{2.0}$, $S_{6}L_{10}T_{2.5}$ and $S_{6}L_{20}T_{2.0}$, respectively. Similar to the upstream control cases, figure 27 shows that $P_{y}$ always has a positive effect on the growth of symmetric modes. The main difference is that the peak value of $P_{y}$ in the downstream control case is lower than that in the corresponding case with the upstream control effect. This indicates that the downstream wall heating strip has a stronger suppressive effect on the contribution of wall-normal shear to disturbance growth; therefore, the transition is delayed further downstream.

Figure 27. Production term $P_{y}$ of disturbance kinetic energy at (a,d,g) $x-x_{r}=3\delta _{r}$, (b,e,h) $x-x_{r}=6\delta _{r}$ and (c,f,i) $x-x_{r}=9\delta _{r}$ for (a–c) case $S_{6}L_{10}T_{2.0}$, (d–f) case $S_{6}L_{10}T_{2.5}$ and (g–i) case $S_{6}L_{20}T_{2.0}$. The grey solid curves are isolines of the streamwise velocity. The black dashed curve is the outline of the roughness element.

The effects of the spanwise shear are analysed through the distributions of $P_{z}$ in figure 28. Generally, $P_{z}$ has clear negative distributions at $x-x_{r}=3\delta _{r}$. However, the negative peak values gradually decrease as $T_{strip}$ or $L_{strip}$ increases. At the second position of $x-x_{r}=6\delta _{r}$, neither the positive nor negative values are pronounced. This changes at $x-x_{r}=9\delta _{r}$. At this position, $P_{z}$ shows clear position peaks in the separated shear layer region, showing that the spanwise shear layer clearly contributes to the growth of the symmetric mode. Compared with the upstream control cases, the wall-normal shear contributes less to the growth of the symmetric mode in the downstream control cases, while the spanwise shear contributes more. This indicates that the upstream wall heating strip weakens the wake vortices and the separation shear layer as a whole; thus, the contributions of the wall-normal and spanwise shear are both weakened. By contrast, the downstream wall heating strip mainly wakens the wall-normal component of the local shear layer; thus, $P_{y}$ is weakened significantly, but $P_{z}$ is less influenced. However, the peak positive value of $P_{y}$ is much greater than that of $P_{z}$; therefore, the contribution of wall-normal shear is more important.

Figure 28. Production term $P_{z}$ of disturbance kinetic energy at (a,d,g) $x-x_{r}=3\delta _{r}$, (b,e,h) $x-x_{r}=6\delta _{r}$ and (c,f,i) $x-x_{r}=9\delta _{r}$ for (a–c) case $S_{6}L_{10}T_{2.0}$, (d–f) case $S_{6}L_{10}T_{2.5}$ and (g–i) case $S_{6}L_{20}T_{2.0}$. The grey solid curves are isolines of the streamwise velocity. The black dashed curve is the outline of the roughness element.

5.3. Combination of upstream and downstream controls

This section further investigates the transition control effect through the combination of upstream and downstream wall heating strips for cases $S_{\pm 6}L_{10}T_{2.0}$ and $S_{\pm 6}L_{20}T_{2.0}$. Figure 29 first plots the instantaneous streamwise velocity for the two cases. In case $S_{\pm 6}L_{10}T_{2.0}$, the breakdown position is located at $x-x_{r}\approx 46\delta _{r}$, corresponding to a delay of approximately $16\delta _{r}$ and $10\delta _{r}$ compared with cases $S_{-6}L_{10}T_{2.0}$ and $S_{6}L_{10}T_{2.0}$, respectively. In case $S_{\pm 6}L_{10}T_{2.0}$, with two wide control strips, the flow is fully stabilized in the whole computational domain, and no transition is observed. Therefore, we can conclude that the combination of upstream and downstream strips more efficiently suppresses the roughness-induced transition than the use of single upstream or downstream wall heating strips.

Figure 29. Instantaneous streamwise velocity at $y=0.5 k_{r}$ for (a) case $S_{\pm 6}L_{10}T_{2.0}$ and (b) case $S_{\pm 6}L_{20}T_{2.0}$.

The flow structures in the roughness wake region are analysed in figure 30. Generally, with the combined upstream and downstream strips, the horseshoe vortex and counter-rotating vortex pairs become much weaker. Thus, the boundary layer distortion is relaxed, and the separation shear layer becomes weaker. As a result, the interaction between the vortex system and separation shear layer and the increase in the streamwise velocity fluctuations in the wake region are significantly reduced.

Figure 30. Slices of (a,b) case $S_{\pm 6}L_{10}T_{2.0}$ and (c,d) case $S_{\pm 6}L_{20}T_{2.0}$ at $x-x_{r}=10\delta _{r}$ coloured by (a,c) $Q$ criterion and (b,d) streamwise velocity fluctuation. The grey solid curves in (a,c) and (b,d) are isolines of the streamwise velocity and local shear value, respectively. The black dashed curve is the outline of the roughness element.

Furthermore, BiGlobal linear stability analysis is performed at $St=0.462$ and $x-x_{r} =10\delta _{r}$ for the two cases (figure 31). In case $S_{\pm 6}L_{10}T_{2.0}$, symmetric and antisymmetric modes can still be found. Compared with cases $S_{-6}L_{10}T_{2.0}$ and $S_{6}L_{10}T_{2.0}$, the two modes have lower growth rates. In particular, the growth rate of the antisymmetric mode is relatively low and is very close to zero. This means that the antisymmetric mode is nearly fully suppressed in this case. In case $S_{\pm 6}L_{20}T_{2.0}$, only a very weak symmetric mode is observed, and its growth rate is much lower than that in cases $S_{-6}L_{10}T_{2.0}$ and $S_{6}L_{10}T_{2.0}$.

Figure 31. Eigenvalue spectra at $x-x_{r}=10\delta _{r}$ with $St=0.462$ for (a) case $S_{\pm 6}L_{10}T_{2.0}$ and (b) case $S_{\pm 6}L_{20}T_{2.0}$.