18. Potential equation for irrotational compressible flows

Under the assumption of irrotational flow1, the velocity field \(\mathbf{u}(\mathbf{r},t)\) can be written as a gradient of a scalar potential \(\phi(\mathbf{r},t)\),

\[\nabla \times \mathbf{u} = \mathbf{0} \quad \rightarrow \quad \mathbf{u} = \nabla \phi \ .\]

Euler equations in convective form with independent variables \((\rho, \mathbf{u}, s)\) read2

\[\begin{split}\begin{aligned} & \partial_t \rho + \mathbf{u} \cdot \nabla \rho + \rho \nabla \cdot \mathbf{u} = 0 \\ & \partial_t \mathbf{u} + \nabla \frac{|\mathbf{u}|^2}{2} + \frac{\nabla p}{\rho} = \mathbf{0} \\ & \partial_t s + \mathbf{u} \cdot \nabla s = 0 \ . \end{aligned}\end{split}\]

18.1. Homoentropic flows

If no shock occurs and the inflow is uniform, the flow is homoentropic, \(s(\mathbf{r},t) = \overline{s}\). Mass and momentum equations become

\[\begin{split}\begin{aligned} & \partial_t \rho + \nabla \phi \cdot \nabla \rho + \rho \nabla^2 \phi = 0 \\ & \partial_t \nabla \phi + \nabla \frac{|\nabla \phi|^2}{2} + a^2(\rho, \overline{s}) \frac{\nabla \rho}{\rho} = \mathbf{0} \ , \end{aligned}\end{split}\]

being \(a^2(\rho, \overline{s}) = \left( \frac{\partial p}{\partial \rho} \right)_s\) the square of the speed of sound.

Momentum equation

Momentum equation can be manipulated as

\[\begin{split}\begin{aligned} \mathbf{0} & = \partial_t \mathbf{u} + \nabla \frac{|\mathbf{u}|^2}{2} + \frac{\nabla p}{\rho} = \\ & = \partial_t \nabla \phi + \nabla \frac{|\nabla \phi|^2}{2} + \frac{\nabla p}{\rho} = \\ & = \partial_t \nabla \phi + \nabla \frac{|\nabla \phi|^2}{2} + \nabla h(\rho, \overline{s}) = \\ & = \nabla \left[ \partial_t \phi + \frac{|\nabla \phi|^2}{2} + h(\rho, \overline{s}) \right] \ , \end{aligned}\end{split}\]

and thus

\[\partial_t \phi + \frac{|\nabla \phi|^2}{2} + h(\rho, \overline{s}) = C(t) \ ,\]

for homoentropic flows \(\nabla s = 0\), as the relation (todo add link to thermodynamics)

\[d h = T ds + \frac{dp}{\rho} \ ,\]

produces

\[d h = \frac{dp}{\rho} \quad , \quad \nabla h = \frac{\nabla p}{\rho} \ .\]

todo

• Discuss if and why the function \(C(t)\) can be included in the definition of «\(\phi(\mathbf{r},t) = \phi(\mathbf{r},t) + C(t)\)».

• or With homogeneous and steady inflow,

  \[\partial_t \phi(\mathbf{r},t) + \frac{|\nabla \phi(\mathbf{r},t)|^2}{2} + h(\rho(\mathbf{r},t), \overline{s}) = \frac{|\mathbf{u}_\infty|^2}{2} + h_\infty \ .\]

Taking partial derivative in time

\[\begin{split}\begin{aligned} 0 & = \partial_{tt} \phi + \partial_t \frac{|\nabla \phi|^2}{2} + \partial_t h = \\ & = \partial_{tt} \phi + \partial_t \frac{|\nabla \phi|^2}{2} + \left( \frac{\partial h}{\partial \rho} \right)_s \partial_t \rho = \\ & = \partial_{tt} \phi + \partial_t \frac{|\nabla \phi|^2}{2} + \frac{a^2(\rho,\overline{s})}{\rho} \partial_t \rho \ , \end{aligned}\end{split}\]

as

\[\left( \frac{\partial h}{\partial \rho} \right)_s = \frac{1}{\rho} \left( \frac{\partial p}{\partial \rho} \right)_s = \frac{a^2(\rho, \overline{s})}{\rho} \ .\]

\[\begin{split}\begin{aligned} 0 & = \partial_{tt} \phi + \partial_t \frac{|\nabla \phi|^2}{2} + \frac{a^2}{\rho} \partial_t \rho && \left( \text{mass:} \ \partial_t \rho = - \rho \nabla^2 \phi - \nabla \rho \cdot \nabla \phi \right) \\ & = \partial_{tt} \phi + \partial_t \frac{|\nabla \phi|^2}{2} - \frac{a^2}{\rho} \left\{ \nabla \phi \cdot \nabla \rho + \rho \nabla^2 \phi \right\} = && \left( \text{momentum:} \ a^2 \frac{\nabla \rho}{\rho} = - \left( \partial_t \nabla \phi + \nabla \frac{|\nabla \phi|^2}{2} \right) \right) \\ & = \partial_{tt} \phi + \partial_t \frac{|\nabla \phi|^2}{2} - a^2 \nabla^2 \phi + \nabla \phi \cdot \left( \partial_t \nabla \phi + \nabla \frac{|\nabla \phi|^2}{2} \right) \ , \end{aligned}\end{split}\]

or

\[0 = \partial_{tt} \phi - a^2 \nabla^2 \phi + \partial_t |\nabla \phi|^2 + \nabla \phi \cdot \nabla \frac{|\nabla \phi|^2}{2} \ .\]

18.1.1. Linearized equation

Assuming small perturbation of the asymptotic flow, with velocity \(\mathbf{u}_\infty\) and thermodynamic state \(\text{TD}_{\infty}\) with speed of sound \(a^2_\infty\), the velocity field can be written as

\[\mathbf{u}(\mathbf{r},t) \sim \mathbf{u}_\infty + \mathbf{u}'(\mathbf{r},t) \ ,\]

the potential as

\[\phi(\mathbf{r},t) \sim \mathbf{u}_\infty \cdot \mathbf{r} + \varphi(\mathbf{r},t) \ ,\]

and all the thermodynamic variables their linear Taylor expansion around the reference state

\[\begin{split}\begin{aligned} \rho & = \rho_\infty + \rho' \\ & \dots \end{aligned}\end{split}\]

Linearization of the potential equation gives

\[D^{\mathbf{u}_\infty}_{tt} \varphi - a^2_\infty \nabla^2 \varphi = 0 \ ,\]

with the differential operator - here applied twice3 - \(D^{\mathbf{u}_\infty}_t = \partial_t + \mathbf{u}_\infty \cdot \nabla\) representing advection with the free-stream velocity field \(\mathbf{u}_\infty\).

Details

\[\begin{split}\begin{aligned} 0 & = \partial_{tt} \phi - a^2 \nabla^2 \phi + \partial_t |\nabla \phi|^2 + \nabla \phi \cdot \nabla \frac{|\nabla \phi|^2}{2} = \\ & = \partial_{tt} \varphi - ( a^2_\infty + {a'}^2 ) \nabla^2 \varphi + 2 \partial_t ( \nabla \phi \cdot \mathbf{u}_\infty + \dots ) + \mathbf{u}_\infty \cdot \nabla \left( \mathbf{u}_\infty \cdot \nabla \varphi + \dots \right) \simeq \\ & = \partial_{tt} \varphi - a^2_{\infty} \nabla^2 \varphi + 2 \mathbf{u}_\infty \cdot \partial_t \nabla \phi + \mathbf{u}_\infty \cdot \nabla ( \mathbf{u}_\infty \cdot \nabla \varphi ) \ . \end{aligned}\end{split}\]

Introducing the linear differential operator

\[D^{\mathbf{u}_\infty}_t := \partial_t + \mathbf{u}_\infty \ ,\]

the linearized equation can be re-written as

\[D^{\mathbf{u}_\infty}_{tt} \varphi - a^2_\infty \nabla^2 \varphi = 0 \ ,\]

being \(D^{\mathbf{u}_\infty}_{tt}\) the operator applied twice, \(D^{\mathbf{u}_\infty}_{tt} = D^{\mathbf{u}_\infty}_{t}D^{\mathbf{u}_\infty}_{t}\).

Introducing a set of Cartesian coordinates with the \(\hat{\mathbf{x}}\)-axis aligned with the free-stream velocity \(\mathbf{u}_\infty = U_{\infty} \hat{\mathbf{x}}\), the differential operator can be written as

\[D_t^{\mathbf{u}_\infty} = \partial_t + U_\infty \partial_x \ ,\]

and the linearized equation of the potential becomes

\[\varphi_{tt} + 2 U_\infty \partial_{xt} \varphi + U_\infty^2 \varphi_{xx} - a_\infty^2 \left( \varphi_{xx} + \varphi_{yy} + \varphi_{zz} \right) = 0 \ .\]

18.1.2. Linearized steady potential equation - Prandtl-Glauert transformation

Steady equation becomes

\[(1-M_\infty^2) \varphi_{xx} + \varphi_{yy} + \varphi_{zz} = 0 \ .\]

This equation can be transformed into a Laplace equation (the same equation governing the potential for incompressible flows) with a transformation of coordinates

\[\begin{split}\begin{aligned} \widetilde{x} & = \frac{ x }{ \sqrt{1 - M_\infty^2} } \\ \widetilde{y} & = y \\ \widetilde{z} & = z \\ \end{aligned}\end{split}\]

so that

\[\frac{\partial \widetilde{x}}{\partial x} = \frac{1}{ \sqrt{1 - M^2} } \ ,\]

and

\[\varphi_{\widetilde{x} \widetilde{x}} + \varphi_{\widetilde{y} \widetilde{y}} + \varphi_{\widetilde{z} \widetilde{z}} = 0 \ .\]

Details

With

\[\widetilde{x} = \beta x \ ,\]

it follows

\[\frac{\partial}{\partial x} = \frac{\partial \widetilde{x}}{\partial x}\frac{\partial}{\partial \widetilde{x}} = \beta \frac{\partial}{\partial \widetilde{x}} \ ,\]

\[\frac{\partial^2}{\partial x^2} = \beta^2 \frac{\partial^2}{\partial \widetilde{x}^2} \ ,\]

so that

\[\begin{split}\begin{aligned} 0 & = \left( 1 - M_\infty^2 \right) \varphi_{xx} + \varphi_{yy} + \varphi_{zz} = \\ & = \left( 1 - M_\infty^2 \right) \beta^2 \varphi_{\widetilde{x} \widetilde{x}} + \varphi_{yy} + \varphi_{zz} = \\ & = \varphi_{\widetilde{x} \widetilde{x}} + \varphi_{yy} + \varphi_{zz} \ , \end{aligned}\end{split}\]

if \(\beta = \frac{1}{\sqrt{1-M^2_\infty}}\).

18.1.2.1. Nature of the equation

Discussion with 2-dimensional equation first

• If the free-stream is subsonic, \(M_\infty \in [0,1)\)4, the differential equation is elliptic

• If the free-stream is supersonic, \(M_\infty > 1\) (and no shock occurs? Or how to treat them? How strong can they be?) the differential equation is hyperbolic. Method of characteristics…

todo 3-dimensional equation

18.1.3. Pressure coefficient

Pressure coefficient is defined as

\[c_p = \frac{p-p_\infty}{\frac{1}{2}\rho_\infty U_\infty^2}\]

Subsonic flow. Linearizing momentum equation for homoentropic flows,

\[\begin{aligned} h - h_\infty = \frac{U_\infty^2}{2} - \frac{(\mathbf{u}_\infty + \mathbf{u}') \cdot (\mathbf{u}_\infty + \mathbf{u}')}{2} \simeq - \mathbf{u}_\infty \cdot \mathbf{u}' \ . \end{aligned}\]

Using \(dh = T ds + \frac{dp}{\rho}\) for homoentropic flows, \(ds = 0\), \(dp = \rho dh\),

\[p' = p - p_\infty \simeq \rho_\infty ( h' - h_\infty ) = - \rho_\infty \mathbf{u}_\infty \cdot \mathbf{u}' \ ,\]

so that the linearized pressure coefficient reads

\[c_p = - \frac{2 \mathbf{u}_\infty \cdot \mathbf{u}'}{U_\infty^2} \ .\]

With the assumption of \(\mathbf{u}_\infty = U_\infty \hat{\mathbf{x}}\),

\[c_p = - \frac{2}{U_\infty} \frac{\partial \varphi}{\partial x} \ .\]

Now, the pressure coefficient for compressible flow can be compared with the pressure coefficient for incompressible flow \(c_p^{inc}\) in the same domain (e.g. with the same geometry of solid bodies).

\[\begin{split}\begin{aligned} c_p(x,y,z) & = - \frac{2}{U_\infty} \frac{\partial \varphi(x,y,z)}{\partial x} = && \widetilde{\varphi}(\widetilde{x},y,z) = \varphi(x(\widetilde{x}),y,z) \\ & = - \frac{2}{U_\infty} \frac{\partial \widetilde{x}}{\partial x} \frac{\partial \widetilde{\varphi}(\widetilde{x},y,z)}{\partial \widetilde{x}} = \\ & = - \frac{1}{\sqrt{1 - M^2_{\infty}}} \frac{2}{U_\infty} \frac{\partial \widetilde{\varphi}}{\partial \widetilde{x}}(\widetilde{x},y,z \ , \end{aligned}\end{split}\]

so that

\[c_p(x,y,z) = \frac{c_p^{inc}(\widetilde{x},y,z)}{\sqrt{1-M_\infty^2}} \ .\]

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1

Vorticity is confined in thin layers, like boundary layers and wakes, modelled as tangential velocity discontinuities.

2

\(\mathbf{u} \cdot \nabla \mathbf{u} = \nabla \frac{|\mathbf{u}|^2}{2} + \boldsymbol\omega \times \mathbf{u}\), with \(\boldsymbol = \nabla \times \mathbf{u} = \mathbf{0}\) in irrotational flows.

3

\(D^{\mathbf{u}_\infty}_{tt} \varphi := D^{\mathbf{u}_\infty}_{t} \left( D^{\mathbf{u}_\infty}_{t} \varphi \right)\) \(= (\partial_t + \mathbf{u}_\infty \cdot \nabla) (\partial_t \varphi + \mathbf{u}_\infty \cdot \nabla \varphi)\) \(= \partial_{tt} \varphi + 2 \mathbf{u} \cdot \partial_t \nabla \varphi + \mathbf{u}_\infty \cdot \nabla ( \mathbf{u}_\infty \cdot \nabla \varphi)\).

4

In order to get a subsonic flow in the whole domain, the free-stream velocity should be smaller than the speed of sound enough to avoid local supersonic regions in the flow.