This post presents a derivation for the Prandtl–Glauert compressibility correction. The end result is a scaling factor which allows low-speed measurements of aerodynamic forces to be applied at high subsonic Mach numbers. The correction facilitated the design of faster aeroplanes in the 1920s and 1930s, culminating with the peak of piston-engined fighters in World War II.

I was not impressed by the other expositions I found on-line, so this is my contribution. It is based on my scribbled notes from lectures by Jerome Jarret at the University of Cambridge.

We begin with the equations of motion, combine them, and assume isentropic flow to obtain a single equation in velocity only. Next, we assume the velocity field is composed of small irrotational disturbances to a uniform flow, and obtain an equation satisfied by the velocity potential. Finally, we seek a transformation to simplify the problem; with the correct scaling factor, the problem reduces to Laplace’s Equation, the classical result for incompressible flows. Evaluating the aerodynamic forces as a function of the transformed solution reveals the Prandtl–Glauert correction.

Equations of motion

Conservation of mass requires that the divergence of mass flux is zero. Expanding the dot product,

$$ \renewcommand{\vec}[1]{\mathbf{#1}} \renewcommand{\Ma}{\mathit{M\kern-.1ema}} \nabla \cdot (\rho \vec{V}) = 0 \quad \Rightarrow \quad \rho (\nabla \cdot \vec{V}) + \vec{V} \cdot \nabla \rho = 0 .\tag{I} $$

Conservation of momentum for an inviscid fluid in steady flow is described by the Euler Equations,

$$ \rho(\vec{V}\cdot \nabla )\vec{V} = -\nabla p . \tag{II} $$

We now eliminate \(\rho\) and \(\nabla p\) from Equations (I) and (II). For isentropic flow of a perfect gas,

$$ p \propto \rho^\gamma \quad \Rightarrow \quad \left.\frac{\mathrm{d} p}{\mathrm{d} \rho}\right|_s = \frac{p\gamma}{\rho} =\gamma R T = a^2 . $$

Now by the Chain Rule \( \nabla p = a^2 \nabla \rho \) and Equation (I) becomes,

$$ \rho (\nabla \cdot \vec{V}) + \frac{1}{a^2} \vec{V} \cdot \nabla p = 0 , $$

then substituting for \(\nabla p\) using Equation (II) and multiplying by \(a^2/\rho\) yields,

$$ a^2 \nabla \cdot \vec{V} - \vec{V} \cdot (\vec{V} \cdot \nabla)\vec{V} = 0 . \tag{III} $$

This is the governing equation for velocity field in an isentropic, steady, compressible flow of a perfect gas.

Velocity potential form

We now make three additional simplifying assumptions to ease solution of Equation (III):

• Irrotational velocity field;
• Two-dimensional flow;
• Velocity can be expressed as a small perturbation over a uniform background flow.

In an inviscid flow vorticity is convected along streamlines, and with a uniform upstream condition, this means that the entire velocity field is irrotational (in the limit of high Reynolds number, we neglect small viscosity-affected regions at the aerofoil surface). If the velocity field is irrotational, we have \(\nabla \times \vec{V} = 0 \) and there exists a velocity potential \(\psi\) such that \(\vec{V} = \nabla \psi\).

We model flow around a two-dimensional wing in the \(x\)–\(y\) plane as a potential disturbance \(\phi\) to a uniform background flow in the \(x\) direction. The velocity potential is then expressed as,

$$ \psi = V_\infty x + \phi , $$

and from \(\vec{V} = \nabla \psi\) the corresponding velocity vector is,

$$ \vec{V} = \left(V_\infty + \frac{\partial \phi}{\partial x}\right)\vec{i} + \frac{\partial \phi}{\partial y} \vec{j} ,\tag{IV} $$

where \(\vec{i}\) and \(\vec{j}\) are unit vectors in the Cartesian coordinate directions.

Now to find the governing equation on \(\phi\), we substitute Equation (IV) into Equation (III):

$$ \def\Ma{\mathit{M\kern-.1ema}} a^2 \left( \frac{\partial^2 \phi}{\partial x^2} + \frac{\partial^2 \phi}{\partial y^2} \right) - \left( V_\infty +\frac{\partial \phi}{\partial x} \right)^2\frac{\partial^2 \phi}{\partial x^2} \left(\frac{\partial \phi}{\partial y}\right)^2 \frac{\partial^2 \phi}{\partial y^2}=0 , $$

then dividing through by \(a^2\), collecting terms, and using \(\Ma_\infty = V_\infty/a\),

$$ \left[ 1 - \Ma_\infty^2 \left( 1 +\frac{1}{V_\infty}\frac{\partial \phi}{\partial x} \right)^2\right] \frac{\partial^2 \phi}{\partial x^2} + \left[ 1 - \Ma_\infty^2\left( \frac{1}{V_\infty}\frac{\partial \phi}{\partial y} \right)^2 \right] \frac{\partial^2 \phi}{\partial y^2} =0 . $$

Now if we restrict velocity perturbations to be small compared to the free-stream velocity,

$$ \frac{1}{V_\infty}\frac{\partial \phi}{\partial x} \ll 1 ,\frac{1}{V_\infty} \frac{\partial \phi}{\partial y} \ll 1 , $$

and we end up with,

$$ \left( 1 - \Ma_\infty^2 \right) \frac{\partial^2 \phi}{\partial x^2} + \frac{\partial^2 \phi}{\partial y^2} =0 . \tag{V} $$

This is the governing equation for small potential disturbances to a two-dimensional uniform flow. If \(\Ma_\infty = 0 \) this reduces to the familiar incompressible case, where the velocity potential satisfies Laplace’s Equation.

Boundary conditions

We now turn to the boundary conditions that apply to Equation (V). Physically, we want the velocity vector to be tangential to solid walls. To apply this boundary condition we parametrise the position vector on the aerofoil surface according to,

$$ \vec{x}_\mathrm{s}(x) = x \vec{i} + \tau g(x/c) \vec{j} , $$

where the upper and lower surfaces are given by different dimensionless shape functions \(g\) defined over the axial chord fraction \(x/c\), multiplied by a a scalar thickness \(\tau\). A vector normal to this surface is given by,

$$ \vec{n}(x) = - \frac{\mathrm{d}}{\mathrm{d} x}\Big(\tau g(x/c)\Big) \vec{i} + \frac{\mathrm{d}}{\mathrm{d} x}\Big(x\Big) \vec{j} = \frac{-\tau}{c}g’(x/c) \vec{i} + \vec{j} . \tag{VI} $$

Stated mathematically, a tangential velocity vector requires that on the aerofoil surface,

$$ \vec{V} \cdot \vec{n} = 0 , $$

which after substituting Equations (IV) and (VI), and dividing through by \(V_\infty\) becomes,

$$ \left(1 + \frac{1}{V_\infty}\frac{\partial \phi}{\partial x}\right)\left(\frac{-\tau}{c}g’(x/c)\right) + \frac{1}{V_\infty}\frac{\partial \phi}{\partial y} = 0 , $$

and for small perturbations,

$$ \frac{-\tau}{c}g’(x/c)+\frac{1}{V_\infty}\frac{\partial \phi}{\partial y} = 0 . \tag{VII} $$

Transformation

We now seek a scaling of the independent variables in Equation (V) that reduces the number of parameters. If we can arrive at an equation involving only non-dimensional quantities, then it holds for all dimensional flow conditions once properly scaled. Choosing to normalise the coordinates by the chord length,

$$ \hat{x} = x/c , \quad \hat{y} = \beta y / c , $$

where \(\beta\) is a factor to be determined. Substituting this choice into Equation (VII) and rearranging,

$$ \frac{\partial \phi}{\partial \hat{y}} = \frac{\tau V_\infty}{\beta} g’(\hat{x}) . $$

By inspection, we can eliminate the dimensional factor by taking,

$$ \hat{\phi} = \phi \frac{\beta}{\tau V_\infty} \quad \Rightarrow \quad \frac{\partial \hat{\phi}}{\partial \hat{y}} = g’(\hat{x}) . $$

Putting these scalings into Equation (V),

$$ \left( 1 - \Ma_\infty^2 \right)\frac{1}{\beta^2} \frac{\partial^2 \hat{\phi}}{\partial \hat{x}^2} + \frac{\partial^2 \hat{\phi}}{\partial \hat{y}^2} =0 . $$

Finally, if we take \(\beta = \sqrt{1-\Ma_\infty^2}\), then the transformed problem becomes,

$$ \frac{\partial^2 \hat{\phi}}{\partial \hat{x}^2} + \frac{\partial^2 \hat{\phi}}{\partial \hat{y}^2} =0\quad \text{with} \quad \frac{\partial \hat{\phi}}{\partial \hat{y}} = g’(\hat{x})\ \text{on}\ \hat{\vec{x}}_\mathrm{s} .\tag{VIII} $$

This is exactly the same as the incompressible problem at low Mach number. Mach number does not appear in the equation governing \(\hat{\phi}\), so under our assumptions the solutions for \(\hat{\phi}\) are independent of Mach number and identical to the incompressible problem. All dependence on Mach number is captured in the \(\beta\) scaling factor on \(\hat{y}\).

Lift coefficients

To quantify the effect of the scaling on aerodynamic forces like lift, we now need an expression for static pressure as a function of the velocity potential. Substituting Equation (IV) into (II),

$$ -\nabla p = \rho\left(V_\infty +\frac{\partial \phi}{\partial x}\right) \frac{\partial^2 \phi}{\partial x^2} \vec{i} +\rho\frac{\partial \phi}{\partial y} \frac{\partial^2 \phi}{\partial y^2} \vec{j} . $$

Neglect second order terms and integrate with respect to \(x\) from the free-stream to the wall,

$$ \nabla p = -\rho V_\infty \frac{\partial^2 \phi}{\partial x^2} \vec{i}
\quad \Rightarrow \quad p - p_\infty = -\rho_\infty V_\infty \frac{\partial \phi}{\partial x} . $$

The non-dimensional pressure coefficient is then,

$$ C_p = \frac{p - p_\infty}{\frac{1}{2}\rho_\infty V_\infty^2} = \frac{-2}{V_\infty}\frac{\partial \phi}{\partial x} = \frac{-2}{\sqrt{1-\Ma_\infty^2}}\frac{\tau}{c}\frac{\partial \hat{\phi}}{\partial \hat{x}} . \tag{IX} $$

Equation (IX) is powerful. It shows that the non-dimensional static pressure, and hence lift coefficient, is proportional to \(\tau/c\) and inversely proportional to \(\sqrt{1 - \Ma_\infty}\). So if a lift coefficient is known at low Mach number, giving the solution to the incompressible problem for \({\partial \hat{\phi}}/{\partial \hat{x}}\), it can be scaled to be used at higher Mach numbers and different thicknesses. This is the Prandtl–Glauert correction.

The validity of this correction is limited by two of our assumptions. For velocity disturbances to be small with respect to the free-stream velocity, we implicitly require \(\tau/c\ll 1\), so aerofoils cannot be too thick. Also, for Mach numbers approaching unity, shock waves will be present in the flow, invalidating the isentropic flow assumption behind Equation (III).