Nonlinear supersonic indicial response of diamond airfoils

Abstract

Supersonic flows around diamond airfoils, under large-amplitude step motion, are analytically studied. Nine regions with distinct flow structures are identified on each side. These regions include intricate wave structures and significantly shape the indicial response. The novelty of this work is to develop theoretical models, capable of describing each region’s wave speed, pressure, and force. We achieve this by correcting the linear solution to the problem and incorporating nonlinear effects of large angle and airfoil thickness. Our model reports good accuracy regarding the computational fluid dynamics. And three stages, which are predicted by this model, precisely capture important features of aerodynamic force evolution.

Appendix: Solutions to the uniform regions in the nonlinear situation

1. Oblique shock wave Using oblique shock relation, we can get

$$\begin{aligned} \left\{ \begin{array}{l} \tan \alpha _{d}=2\cot \beta \frac{M_{A}^{2}\sin ^{2}\beta -1}{ M_{A}^{2}\left( \gamma +\cos 2\beta \right) +2} \\ \rho _{B}=\rho _{A}\frac{\left( \gamma +1\right) M_{A}^{2}\sin ^{2}\beta }{ 2+\left( \gamma -1\right) M_{A}^{2}\sin ^{2}\beta } \\ \left( M_{B}\right) ^{2}=\frac{M_{A}^{2}+\frac{2}{\gamma -1}}{\frac{2\gamma }{\gamma -1}M_{A}^{2}\sin ^{2}\beta -1}+\frac{M_{A}^{2}\cos ^{2}\beta }{ \frac{\gamma -1}{2}M_{A}^{2}\sin ^{2}\beta +1} \\ u_{B}=M_{B}\times a_{B}\text {, }a_{B}=\sqrt{\frac{\gamma p_{B}}{\rho _{B}}} \\ p_{B}=p_{A}\left( 1+\frac{2\gamma }{\gamma +1}\left( M_{A}^{2}\sin ^{2}\beta -1\right) \right) \\ Cp_{B}=\frac{2}{\gamma M_{\infty }^{2}}\left( \frac{p_{A}}{p_{\infty }} \left( 1+\frac{2\gamma }{\gamma +1}\left( M_{A}^{2}\sin ^{2}\beta -1\right) \right) -1\right) . \end{array} \right. . \end{aligned}$$

(A.1)

Here $\beta$ is the shock angle with respect to $\alpha _{d}$ and $M_{\infty }$.

2. Unsteady normal shock wave We use $M_{S}$ to denote the Mach number of the shock wave. In the step motion, behind unsteady waves are airfoil surfaces. Thus, with $v_{B,n}=0$, the classical solution gives:

$$\begin{aligned} \left\{ \begin{array}{l} M_{S}=\frac{\gamma +1}{4}M_{A}\sin \alpha _{r} +\sqrt{1+\left( \frac{\gamma +1 }{4}M_{A}\sin \alpha _{r} \right) ^{2}} \\ \rho _{B}=\rho _{A}\frac{M_{S}^{2}}{1+\frac{\gamma -1}{\gamma +1}\left( M_{S}^{2}-1\right) } \\ u_{B}=u_{A} \cos \alpha _{r} \text {, }ra_{B}=\sqrt{\frac{\gamma p_{B}}{ \rho _{B}}} \\ p_{B}=p_{\infty } \left( 1+\frac{2\gamma }{\gamma +1}\left( M_{S}^{2}-1 \right) \right) \\ Cp_{B}=\frac{2}{\gamma M_{\infty }^{2}}\left( \frac{p_{A}}{p_{\infty }} \left( 1+\frac{\gamma \left( \gamma +1\right) }{4}M_{A}^{2}\sin ^{2}\alpha _{r}+\gamma \sqrt{M_{A}^{2}\sin ^{2}\alpha _{d}+\left( \frac{\gamma +1}{4} M_{A}^{2}\sin ^{2}\alpha _{d}\right) ^{2}}\right) -1 \right) . \end{array} \right. . \end{aligned}$$

(A.2)

3. Prandtl–Meyer wave Prandtl–Meyer relations give:

$$\begin{aligned} \left\{ \begin{array}{l} \nu (M_{B})-\nu (M_{A})=\alpha _{d} \\ \nu (M)=\sqrt{\frac{\gamma +1}{\gamma -1}}\arctan \left( \sqrt{\frac{\gamma -1}{\gamma +1}}\sqrt{M^{2}-1}\right) -\arctan \sqrt{M^{2}-1} \\ u_{B}=M_{B}\times a_{B}\text {, }a_{B}=\sqrt{\frac{\gamma p_{B}}{\rho _{B}}} \\ \rho _{B}=\rho _{A}\left( \frac{1+\frac{1}{2}\left( \gamma -1\right) M_{A}^{2}}{1+\frac{1}{2}\left( \gamma -1\right) M_{B}^{2}}\right) ^{\frac{1}{ \gamma -1}} \\ p_{B}=p_{A}\left( \frac{1+\frac{1}{2}\left( \gamma -1\right) M_{A}^{2}}{1+ \frac{1}{2}\left( \gamma -1\right) M_{B}^{2}}\right) ^{\frac{\gamma }{\gamma -1}}\ \\ C_{p}=\frac{2\gamma }{M_{\infty }^{2}}\left( \frac{p_{A}}{p_{\infty }}\left( \frac{1+\frac{1}{2}\left( \gamma -1\right) M_{A}^{2}}{1+\frac{1}{2}\left( \gamma -1\right) M_{B}^{2}}\right) ^{\frac{\gamma }{\gamma -1}}-1\right) . \end{array} \right. \end{aligned}$$

(A.3)

4. Unsteady rarefaction wave Considering $v_{B,n}=0$, the solution is given by isentropic relation:

$$\begin{aligned} \left\{ \begin{array}{l} a_{B}=a_{A}-\frac{\gamma -1}{2}V_{A}\sin \alpha _{d} \\ \rho _{B}=\rho _{A}\left( 1-\frac{\gamma -1}{2}M_{A}\sin \alpha _{d}\right) ^{\frac{2}{\gamma -1}} \\ p_{B}=p_{A}\left( 1-\frac{\gamma -1}{2}M_{A}\sin \alpha _{d}\right) ^{\frac{ 2\gamma }{\gamma -1}} \\ C_{p}=\frac{2}{\gamma M_{\infty }^{2}}\left( \frac{p_{A}}{p_{\infty }}\left( 1-\frac{\gamma -1}{2}M_{A}\sin \alpha _{d}\right) ^{\frac{2\gamma }{\gamma -1 }}-1\right) . \end{array} \right. . \end{aligned}$$

(A.4)

For plates, the flow structure and flow parameters are given by Table 7. For any uniform region of the flat plate, one can obtain its solution by taking upstream flow parameters into the corresponding nonlinear wave relation.

Table 7 Flat plate