LES wall modeling for heat transfer at high speeds

A practical application of universal wall scalings is near-wall turbulence modeling. In this paper, we exploit the semilocal scaling [Patel, Boersma, and Pecnik, Phys. Rev. Fluids, 2, 084604 (2017)] and derive an eddy conductivity closure for wall-modeled large-eddy simulation of high-speed flows. We show that while the semilocal scaling does not collapse high-speed direct numerical simulation (DNS) data, the resulting eddy conductivity and the wall model work fairly well. The paper attempts to answer the following outstanding question: why the semilocal scaling fails but the resulting eddy conductivity works well. We conduct DNSs of Couette flows at Mach numbers from $M=1.4$ to 6. We add a source term in the energy equation to get a cold wall, a close-to-adiabatic wall, and a hot wall. Detailed analysis of the flows' energy budgets shows that aerodynamic heating is the answer to our question: Aerodynamic heating is not accounted for in Patel et al.'s semilocal scaling but is modeled in the equilibrium wall model. We incorporate aerodynamic heating in the semilocal scaling and show that the new scaling successfully collapses the high-speed DNS data. We also show that incorporating aerodynamic heating or not, the semilocal scaling gives rise to the exact same eddy conductivity, thereby answering the outstanding question.

Figure 1

A sketch of WMLES. The wall model is a stress-based one.

Figure 2

(a) ${\theta }_T$ as a function of $y^{*}$ for incompressible channel flow at ${\mathrm{Re}}_{\tau }=548$, 995, 2017, and 4088 and for $\mathrm{Pr}=0.7$ and 1.0 [45]. The two thin black lines are ${\theta }_T=y^{*}$ and ${\theta }_T=1/{\kappa }_{T}log\left(y^{*}\right)+B^{\prime }$. For incompressible flows, ${\theta }^{*}=\theta /{\theta }_{\tau }, y^{*}=y^{+}$. (b) $D_T=[{(d{\theta }_T/d{y}^{*})}^{-1}-1]/\left(\kappa y^{*}\right)$ for $y/\delta <0.2$, i.e., below the logarithmic layer. The thin black line corresponds to $D_T\left(y\right)={[1-exp(-y/A_T)]}^2$, where $A_T=20$.

Figure 3

${\theta }_T$ as a function of $y^{*}$ for our DNSs: D3-05C, D3-05H, D3-10C, and D6-05C, Zhang et al.'s DNSs: M6Tw025, M6Tw076, M8Tw048, and M14Tw018 [41], and Coleman et al.'s DNSs: M1.5 and M3. Table 1 tabulates our DNSs' details. The nomenclature of Zhang et al.'s DNSs is as follows: M[A]Tw[B], where A is the freestream Mach number, $B=T_w/T_r\times 100$, and $T_r$ is the recovery temperature. The nomenclature of Coleman et al.'s DNSs is as follows: M[A], where A is the bulk Mach number [44]. The two thin black lines are ${\theta }_T=y^{*}$ and ${\theta }_T=1/{\kappa }_{T}log\left(y^{*}\right)+B^{\prime }$.

Figure 4

Turbulent Prandtl number model. Here the solid blank line and the red dashed line correspond to Eqs. (17) and (19). $\epsilon ={10}^{-6}$ in Eq. (19).

Figure 5

A sketch of the flow configuration. $x, y$, and $z$ are the streamwise, wall-normal, and spanwise coordinates. The flow is periodic in the streamwise and the spanwise directions. The two walls are at the same temperature $T_w$. The wall velocity is $U_w$. $\delta$ is the half channel height.

Figure 6

Viscous and turbulent momentum fluxes as a function of $y/\delta$ in (a) Case D3-05H and (b) Case D3-10C. “Mom. Bud.” denotes “momentum budget.” The thin black line is the sum of the two flux.

Figure 7

Wall model results. The temperature profiles are symmetric with respect to the channel centerline, and we show only the results in the lower half channel.

Figure 8

Error in ${\mathrm{EWM}}^{*}$ for all DNS conditions and for $h_{\mathrm{wm}}=0.01\delta , 0.05\delta , 0.10\delta$, and $0.25\delta$. The two dashed lines are at $\pm 0.025$.

Figure 9

(a) $T/T_w$ as a function of $y/\delta$ for the channel flow DNS in Ref. [43] and for ${\mathrm{WMLES}}^{*}, {\mathrm{WMLES}}_{\mathrm{YL}18}$, and ${\mathrm{WMLES}}^{+}$. The flow condition is ${\mathrm{Re}}_b=34000, M=0.8$. (b) Same as (a) but for $U/a_w$. (c) Same as (a) but at the flow condition ${\mathrm{Re}}_b=34000, M=1.5$. (d) Same as (c) but for $U/a_w$.

Figure 10

$T/T_{\mathrm{ref}}$ as a function of $y/\delta$ at the flow condition (a) 3-05C, (b) 6-05C, and (c) 3-10C.

Figure 11

$T/T_{\mathrm{ref}}$ as a function of $y/\delta$ at the flow condition (a) 3-05A and (c) 3-05H. $(U-U_w)/a_w$ as a function of $y/\delta$ at the flow condition (b) 3-05A and (d) 3-05H.

Figure 12

(a) Turbulent Prandtl number (b) mean velocity, temperature, Reynolds shear stress in ${\mathrm{Re}}_{\tau }=4000$ channel at different thermal conditions [45]. The reader is directed to Ref. [45] for more details of the data.

Figure 13

Energy budget terms in (a) D3-05H, where the heat source term is nonzero, and (b) D3-10C, where the heat source term is $\phi =0$.

Figure 14

Same as Fig. 3 but for the scaling in Eqs. (34) and (6b), which accounts for aerodynamic heating.