Approximating Sensitivities Using Low-Order Models

The core idea underpinning PySAGAS is to use low-order, differentiable fluid models to evaluate the sensitivities of surface pressures to geometric parameters.

High-Level Overview of Approach

Tip

Refer to the nomenclature for definitions of symbols used here.

Vertex-Parameter Sensitivities

To make the link between cell data and geometric parameters, PySAGAS requires the user to provide the vertex-parameter sensitivities, \(\partial\underline{v}/\partial\underline{\theta}\). In simple cases, such as the inclined ramp example, \(\partial\underline{v}/\partial\underline{\theta}\) can be found manually. However, for more complex geometries, this is not possible. For this reason, a parametric geometry generation tool offering geometric differentiation is recommended. One such tool is hypervehicle.

Nominal Surface Properties

PySAGAS also requires the nominal surface properties of the geometry being considered. Namely, the local Mach number, pressure and temperature. This information can come from any source, making PySAGAS CFD-solver agnostic; whether it be Oblique Shock Theory, or Cart3D.

Pressure Sensitivities

Given the nominal surface properties, PySAGAS can calculate the sensitivity of local pressure to changes in the normal vector of a cell. Once these sensitivities are known, they can be linked to the geometric design parameters via application of the chain rule.

Let’s start with piston theory. Surface pressure is given by the following expression, relative to the freestream properties, denoted by the \(\infty\) subscript.

\[ P = P_\infty \, \left( 1 + \frac{\gamma - 1}{2} \, \frac{W}{a_\infty} \right)^{\frac{2 \, \gamma}{\gamma - 1}} \]

In this expression, \(W\) is the downwash speed, \(a_{\infty}\) is the freestream speed of sound, and \(\gamma\) is the ratio of specific heats. By following the approach presented in Zhan (2009)[1], keeping only first-order terms and decomposing the downwash velocity, we obtain “local piston theory”:

\[ P = P_l + \rho_l \, a_l \, W\]

\[ W = \underline{v}_l \cdot \delta \underline{n} + \underline{V}_b \cdot \underline{n} \]

\[ \delta \underline{n} = \underline{n}_0 - \underline{n} \]

Here, the subscript \(l\) refers to the local surface conditions - i.e. the nominal surface properties as described above. In this model, we are only interested in the first term, \(\underline{v}_l \cdot \delta \underline{n}\), as this relates to geometric deformation. The second term, \(\underline{V}_b \cdot \underline{n}\), relates to vibration, and can therefore be discarded. We therefore reach an expression for the local pressure, relative to a cell’s normal vector.

\[ P = P_l + \rho_l \, a_l \, \underline{v}_l \cdot \left( \underline{n}_0 - \underline{n} \right)\]

This expression can be differentiated with respect to the geometric parameters \(\underline{p}\), yielding the sensitivity of pressure to parameters \(\underline{p}\):

\[ \frac{\partial \, P}{\partial \, \underline{\theta}} = \rho_l \, a_l \, \underline{v}_l \cdot \left(- \frac{\partial \, \underline{n}}{\partial \, \underline{\theta}} \right) \]

Of course, this expression requires \(d\underline{n}/d\underline{p}\). Conveniently, this can be evaluated using the vertex parameter sensitivities and the normal vector vertex sensitivities as follows:

\[ \frac{\partial \, \underline{n}}{\partial \, \underline{\theta}} = \frac{\partial \, \underline{n}}{\partial \, \underline{v}} \frac{\partial \, \underline{v}}{\partial \, \underline{\theta}} \]

Aerodynamic Sensitivities

With the sensitivity of pressure to parameters, \(\partial P/ \partial \underline{\theta}\), PySAGAS is able to calculate the sensitivity of the aerodynamic surface properties to the geometric design parameters (i.e. \(\frac{\partial \, C_X}{\partial \, \underline{\theta}}\)). First, note that the force acting on a given cell is the product of the pressure acting on that cell, and the cell’s area. The total force acting on the geometry is therefore the sum of cell forces:

\[ \underline{F}_{\text{ i}} = P_i(\underline{\theta}) \, A_i(\underline{\theta}) \, \underline{n}_{i}(\underline{\theta}) \]

\[ \underline{F}_{\text{ net}} = \sum_{i=0}^{N} \underline{F}_{\text{ i}} \]

This expression can be differentiated with respect to the geometric parameters to yield the following.

\[ \frac{\partial\underline{F}_{\text{net}}}{\partial\underline{\theta}_j} = \sum_{i=0}^N \left[ \underbrace{ \frac{\partial P_i}{\partial\underline{\theta}_j} A_i(\underline{\theta}_j)\underline{n}_i(\underline{\theta}_j) }_\text{pressure sensitivity} \: + \: \underbrace{ P_i(\underline{\theta}_j) \frac{\partial A_i}{\partial\underline{\theta}_j} \underline{n}_i(\underline{\theta}_j) }_\text{area sensitivity} \: + \: \underbrace{ P_i(\underline{\theta}_j) A_i(\underline{\theta}_j)\frac{\partial\underline{n}_i}{\partial\underline{\theta}_j} }_\text{normal sensitivity} \right] \]

These sensitivities can also be converted into non-dimensional force coefficients by dividing through by \(\frac{1}{2} \, \rho_\infty \, V_\infty^2 \, A_{ref} \).