Interaction theory of hypersonic laminar near-wake flow behind an adiabatic circular cylinder

Abstract

The separation and shock wave formation on the aft-body of a hypersonic adiabatic circular cylinder were studied numerically using the open source software OpenFOAM. The simulations of laminar flow were performed over a range of Reynolds numbers ($8\times 10^3 < Re < 8\times 10^4$) at a free-stream Mach number of 5.9. Off-body viscous forces were isolated by controlling the wall boundary condition. It was observed that the off-body viscous forces play a dominant role compared to the boundary layer in displacement of the interaction onset in response to a change in Reynolds number. A modified free-interaction equation and correlation parameter has been presented which accounts for wall curvature effects on the interaction. The free-interaction equation was manipulated to isolate the contribution of the viscous–inviscid interaction to the overall pressure rise and shock formation. Using these equations coupled with high-quality simulation data, the underlying mechanisms resulting in Reynolds number dependence of the lip-shock formation were investigated. A constant value for the interaction parameter representing the part of the pressure rise due to viscous–inviscid interaction has been observed at separation over a wide range of Reynolds numbers. The effect of curvature has been shown to be the primary contributor to the Reynolds number dependence of the free-interaction mechanism at separation. The observations in this work have been discussed here to create a thorough analysis of the Reynolds number-dependent nature of the lip-shock.

Appendix: Derivation of free-interaction for a cylindrical aft-body

The following derivation follows closely the derivation given by Délery [16] for the Chapman [13] free-interaction theory. Starting with the boundary layer momentum equation in the streamwise direction at the wall the following simple relationship can be found. In these derivations, x refers to the streamwise distance, and y refers to the wall normal distance.

$$\begin{aligned} \displaystyle \frac{{\hbox {d}}p}{{\hbox {d}}x}=\left( \frac{\partial \tau }{\partial y}\right) _\mathrm{w} \end{aligned}$$

(16)

Integrating both sides:

$$\begin{aligned} \displaystyle p\left( \bar{x}\right) -p\left( \bar{x}_\mathrm{o}\right) =\int \limits _{x_\mathrm{o}}^{x}\left( \frac{\partial \tau }{\partial y}\right) _\mathrm{w} {\hbox {d}}x \end{aligned}$$

(17)

Introducing the non-dimensional variables $\bar{\tau }=\tau /\tau _\mathrm{wo}$, $\bar{y}=y/\delta _\mathrm{o}^*$, $q_\mathrm{o}=0.5\gamma p_\mathrm{e}\left( x_\mathrm{o}\right) M_\mathrm{e,o}^2$ and $C_{fo}=\tau _\mathrm{wo}/q_\mathrm{o}$ the equation becomes:

$$\begin{aligned} \displaystyle \frac{p\left( \bar{x}\right) -p\left( \bar{x}_\mathrm{o}\right) }{q_\mathrm{o}}=C_{fo}\frac{L_ sep }{\delta _\mathrm{o}^*} f_1\left( \bar{x}\right) \end{aligned}$$

(18)

where:

$$\begin{aligned} \displaystyle f_1\left( \bar{x}\right) =\int \limits _{\bar{x}_\mathrm{o}}^{\bar{x}}\left( \frac{\partial \bar{\tau }}{\partial \bar{y}}\right) _\mathrm{w} {\hbox {d}}\bar{x} \end{aligned}$$

(19)

This is the first equation needed to define the interaction. The second equation represents the pressure rise due to external flow deflection.

$$\begin{aligned} \displaystyle \frac{\sqrt{M_\mathrm{e,o}^2-1}}{\gamma M_\mathrm{e,o}^2} \frac{{\hbox {d}}p}{p}={\hbox {d}} \varphi \end{aligned}$$

(20)

In its linearized form:

$$\begin{aligned} \displaystyle \frac{\sqrt{M_\mathrm{e,o}^2-1}}{\gamma M_\mathrm{e,o}^2} \frac{\Delta p}{p}=\Delta \varphi \end{aligned}$$

(21)

The following is the only significant difference from the derivation of Chapman [13, 16]. The streamline deflection angle $\varphi$ in the case of a cylinder is the sum of the deflection from viscous–inviscid interaction and the deflection of the cylindrical aft-body. As well, because we know that there is vertical pressure gradient and rotational flow, the deflection is actually an effective angle that would result in the given pressure rise. In the general case of flat-plate flow it is a good assumption that the compression is only due the displacement thickness growth ${\hbox {d}} \delta ^* / {\hbox {d}}x$. Here we do not specify that this is the specific viscous–inviscid mechanism. Therefore:

$$\begin{aligned} \displaystyle \frac{\sqrt{M_\mathrm{e,o}^2-1}}{\gamma M_\mathrm{e,o}^2} \frac{\Delta p}{p}=\Delta \varphi _{ eff } + \Delta \alpha , \end{aligned}$$

(22)

where $\Delta \varphi _{ eff }$ is the effective angle change due to viscous–inviscid interaction. $\Delta \alpha$ is the angle change due to the surface deflection of the cylinder. If we assume that $\Delta \varphi _{ eff }$ scales with $\delta _\mathrm{o}^*/L_{ sep }$, and $\Delta \alpha$ scales with the angular length of the interaction $\theta _{ sep }=L_{ sep }/r$ and introducing the same dimensionless variables as the previous step, we can get the following function:

$$\begin{aligned} \displaystyle \frac{p\left( \bar{x}\right) -p\left( \bar{x}_\mathrm{o}\right) }{q_\mathrm{o}}=\frac{2}{\sqrt{M_\mathrm{e,o}^2-1}}\left[ \frac{\delta _\mathrm{o}^*}{L_{ sep }}f_2\left( \bar{x}\right) +\theta _{ sep }f_3\left( \bar{x}\right) \right] \end{aligned}$$

(23)

The function $f_2\left( \bar{x}\right)$ is the dimensionless effective angle change due to viscous–inviscid interaction. In the flat-plate case this can be assumed to be equal to ${\hbox {d}}\bar{\delta _\mathrm{o}^*}/{\hbox {d}}\bar{x}$. The function $f_3\left( \bar{x}\right)$ is the dimensionless angle change due to the cylinder angle. Multiplying (18) and (23) and taking the square root we get the following:

$$\begin{aligned} \frac{p\left( \bar{x}\right) -p\left( \bar{x}_\mathrm{o}\right) }{q_\mathrm{o}}= & {} \sqrt{\frac{2C_{fo}}{\left( M_\mathrm{e,o}^2-1\right) ^{1/2}}}\nonumber \\&\left[ f_1\left( \bar{x}\right) f_2\left( \bar{x}\right) +\frac{L_{ sep }^2}{\delta _\mathrm{o}^*r}f_1\left( \bar{x}\right) f_3\left( \bar{x}\right) \right] ^{0.5}\nonumber \\ \end{aligned}$$

(24)

or

$$\begin{aligned} \displaystyle \frac{p\left( \bar{x}\right) -p\left( \bar{x}_\mathrm{o}\right) }{q_\mathrm{o}}=F^*\left( \bar{x}\right) \sqrt{\frac{2C_{fo}}{\left( M_\mathrm{e,o}^2-1\right) ^{1/2}}} \end{aligned}$$

(25)

The above equation is in the same form as the free-interaction theory equation from Chapman [13]. However, the interaction parameter is denoted $F^*\left( \bar{x}\right)$ because it incorporates both the effect of the geometry and the free-interaction. This equation would also be appropriate for a geometry other than a cylinder provided the radius of curvature is relatively constant throughout the interaction region. Equation 24 can be rearranged for the free-interaction parameter $F\left( \bar{x}\right) =\sqrt{f_1\left( \bar{x}\right) f_2\left( \bar{x}\right) }$ to give:

$$\begin{aligned} \displaystyle F\left( \bar{x}\right)= & {} \sqrt{f_1\left( \bar{x}\right) f_2\left( \bar{x}\right) }\nonumber \\= & {} \left[ \left( \frac{p\left( \bar{x}\right) -p\left( \bar{x}_\mathrm{o}\right) }{q_\mathrm{o}}\right) ^2\frac{\sqrt{M_\mathrm{e,o}^2-1}}{2C_{fo}}-\frac{L_{ sep }^2}{\delta _\mathrm{o}^*r}f_1\left( \bar{x}\right) f_3\left( \bar{x}\right) \right] ^{0.5}\nonumber \\ \end{aligned}$$

(26)