Editing Potential flow (section)

==Compressible flow==
===Steady flow===<!-- [[Full potential equation]] redirects here -->
Potential flow theory can also be used to model irrotational compressible flow. The derivation of the governing equation for <math>\varphi</math> from [[Euler equations (fluid dynamics)|Eulers equation]] is quite straightforward. The continuity and the (potential flow) momentum equations for steady flows are given by

<math display="block">\rho \nabla\cdot\mathbf v + \mathbf v\cdot\nabla \rho = 0, \quad (\mathbf v \cdot\nabla)\mathbf v= -\frac{1}{\rho}\nabla p = -\frac{c^2}{\rho}\nabla \rho</math>

where the last equation follows from the fact that [[entropy]] is constant for a fluid particle and that square of the [[sound speed]] is <math>c^2=(\partial p/\partial\rho)_s</math>. Eliminating <math>\nabla\rho</math> from the two governing equations results in

<math display="block">c^2\nabla\cdot\mathbf v - \mathbf v\cdot (\mathbf v \cdot \nabla)\mathbf v=0.</math>

The incompressible version emerges in the limit <math>c\to\infty</math>. Substituting here <math>\mathbf v=\nabla\varphi</math> results in<ref name="landau">{{cite book | last1 = Landau | first1 = L. D. | last2 = Lifshitz | first2 = E. M. | year = 2013 | title = Fluid mechanics | series = Landau And Lifshitz: Course of Theoretical Physics | volume = 6 | publisher = Elsevier | section = 114 | page = 436}}</ref><ref name=Anderson>{{cite book | first=J. D. | last=Anderson | author-link=John D. Anderson | title=Modern compressible flow | year=2002 | publisher=McGraw-Hill | isbn=0-07-242443-5 |pages= 358–359}}</ref>

<math display="block">(c^2-\varphi_x^2)\varphi_{xx}+(c^2-\varphi_y^2)\varphi_{yy}+(c^2-\varphi_z^2)\varphi_{zz}-2(\varphi_x\varphi_y\varphi_{xy}+\varphi_y\varphi_z\varphi_{yz}+\varphi_z\varphi_x\phi_{zx})=0</math>

where <math>c=c(v)</math> is expressed as a function of the velocity magnitude <math>v^2=(\nabla\phi)^2</math>. For a [[polytropic gas]], <math>c^2 = (\gamma-1)(h_0-v^2/2)</math>, where <math>\gamma</math> is the [[specific heat ratio]] and <math>h_0</math> is the [[stagnation enthalpy]]. In two dimensions, the equation simplifies to 

<math display="block">(c^2-\varphi_x^2)\varphi_{xx}+(c^2-\varphi_y^2)\varphi_{yy}-2\varphi_x\varphi_y\varphi_{xy}=0.</math>

'''Validity:''' As it stands, the equation is valid for any inviscid potential flows, irrespective of whether the flow is subsonic or supersonic (e.g. [[Prandtl–Meyer expansion fan|Prandtl–Meyer flow]]). However in supersonic and also in transonic flows, shock waves can occur which can introduce entropy and vorticity into the flow making the flow rotational. Nevertheless, there are two cases for which potential flow prevails even in the presence of shock waves, which are explained from the (not necessarily potential) momentum equation written in the following form

<math display="block">\nabla (h+v^2/2) - \mathbf v\times\boldsymbol\omega = T \nabla s</math>

where <math>h</math> is the [[specific enthalpy]], <math>\boldsymbol\omega</math> is the [[vorticity]] field, <math>T</math> is the temperature and <math>s</math> is the specific entropy. Since in front of the leading shock wave, we have a potential flow, Bernoulli's equation shows that <math>h+v^2/2</math> is constant, which is also constant across the shock wave ([[Rankine–Hugoniot conditions]]) and therefore we can write{{r|landau}}

<math display="block">\mathbf v\times\boldsymbol\omega = -T \nabla s</math>

1) When the shock wave is of constant intensity, the entropy discontinuity across the shock wave is also constant i.e., <math>\nabla s=0</math> and therefore vorticity production is zero. Shock waves at the pointed leading edge of two-dimensional wedge or three-dimensional cone ([[Taylor–Maccoll flow]]) has constant intensity. 2) For weak shock waves, the entropy jump across the shock wave is a third-order quantity in terms of shock wave strength and therefore <math>\nabla s</math> can be neglected. Shock waves in slender bodies lies nearly parallel to the body and they are weak.

'''Nearly parallel flows:''' When the flow is predominantly unidirectional with small deviations such as in flow past slender bodies, the full equation can be further simplified. Let <math>U\mathbf{e}_x</math> be the mainstream and consider small deviations from this velocity field. The corresponding velocity potential can be written as <math>\varphi = x U + \phi</math> where <math>\phi</math> characterizes the small departure from the uniform flow and satisfies the linearized version of the full equation. This is given by

<math display="block">(1-M^2) \frac{\partial^2\phi}{\partial x^2} + \frac{\partial^2\phi}{\partial y^2} + \frac{\partial^2\phi}{\partial z^2} =0</math>

where <math>M=U/c_\infty</math> is the constant [[Mach number]] corresponding to the uniform flow. This equation is valid provided <math>M</math> is not close to unity. When <math>|M-1|</math> is small (transonic flow), we have the following nonlinear equation{{r|landau}}

<math display="block">2\alpha_*\frac{\partial\phi}{\partial x} \frac{\partial^2\phi}{\partial x^2} = \frac{\partial^2\phi}{\partial y^2} + \frac{\partial^2\phi}{\partial z^2}</math>

where <math>\alpha_*</math> is the critical value of [[Landau derivative]] <math>\alpha = (c^4/2\upsilon^3)(\partial^2 \upsilon/\partial p^2)_s</math><ref>1942, Landau, L.D. "On shock waves" J. Phys. USSR 6 229-230</ref><ref>Thompson, P. A. (1971). A fundamental derivative in gasdynamics. The Physics of Fluids, 14(9), 1843-1849.</ref> and <math>\upsilon=1/\rho</math> is the specific volume. The transonic flow is completely characterized by the single parameter <math>\alpha_*</math>, which for polytropic gas takes the value <math>\alpha_*=\alpha=(\gamma+1)/2</math>. Under [[hodograph]] transformation, the transonic equation in two-dimensions becomes the [[Euler–Tricomi equation]].

=== Unsteady flow ===<!-- [[Full potential equation]] redirects here -->
The continuity and the (potential flow) momentum equations for unsteady flows are given by

<math display="block">\frac{\partial\rho}{\partial t} + \rho \nabla\cdot\mathbf v + \mathbf v\cdot\nabla \rho = 0, \quad \frac{\partial\mathbf v}{\partial t}+ (\mathbf v \cdot\nabla)\mathbf v= -\frac{1}{\rho}\nabla p =-\frac{c^2}{\rho}\nabla \rho=-\nabla h.</math>

The first integral of the (potential flow) momentum equation is given by 

<math display="block">\frac{\partial\varphi}{\partial t} + \frac{v^2}{2} + h = f(t), \quad \Rightarrow \quad \frac{\partial h}{\partial t} = -\frac{\partial^2\varphi}{\partial t^2} - \frac{1}{2}\frac{\partial v^2}{\partial t} + \frac{df}{dt}</math>

where <math>f(t)</math> is an arbitrary function. Without loss of generality, we can set <math>f(t)=0</math> since <math>\varphi</math> is not uniquely defined. Combining these equations, we obtain

<math display="block">\frac{\partial^2\varphi}{\partial t^2} + \frac{\partial v^2}{\partial t}=c^2\nabla\cdot\mathbf v - \mathbf v\cdot (\mathbf v \cdot \nabla)\mathbf v.</math>

Substituting here <math>\mathbf v=\nabla\varphi</math> results in

<math display="block">\varphi_{tt} + (\varphi_x^2+ \varphi_y^2+ \varphi_z^2)_t= (c^2-\varphi_x^2)\varphi_{xx}+(c^2-\varphi_y^2)\varphi_{yy}+(c^2-\varphi_z^2)\varphi_{zz}-2(\varphi_x\varphi_y\varphi_{xy}+\varphi_y\varphi_z\varphi_{yz}+\varphi_z\varphi_x\phi_{zx}).</math>

'''Nearly parallel flows:''' As in before, for nearly parallel flows, we can write (after introudcing a recaled time <math>\tau=c_\infty t</math>)

<math display="block">\frac{\partial^2\phi}{\partial \tau^2}  +  2M \frac{\partial^2\phi}{\partial x\partial\tau}= (1-M^2) \frac{\partial^2\phi}{\partial x^2} + \frac{\partial^2\phi}{\partial y^2} + \frac{\partial^2\phi}{\partial z^2}</math>

provided the constant Mach number <math>M</math> is not close to unity. When <math>|M-1|</math> is small (transonic flow), we have the following nonlinear equation{{r|landau}}

<math display="block">\frac{\partial^2\phi}{\partial \tau^2}  +  2\frac{\partial^2\phi}{\partial x\partial\tau} = -2\alpha_*\frac{\partial\phi}{\partial x} \frac{\partial^2\phi}{\partial x^2} + \frac{\partial^2\phi}{\partial y^2} + \frac{\partial^2\phi}{\partial z^2}.</math>

'''Sound waves:''' In sound waves, the velocity magntiude <math>v</math> (or the Mach number) is very small, although the unsteady term is now comparable to the other leading terms in the equation. Thus neglecting all quadratic and higher-order terms and noting that in the same approximation, <math>c</math> is a constant (for example, in polytropic gas <math>c^2=(\gamma-1)h_0</math>), we have<ref>Lamb (1994) §287, pp. 492–495.</ref>{{r|landau}}

<math display="block">\frac{\partial^2 \varphi}{\partial t^2} = c^2 \nabla^2 \varphi,</math>

which is a linear [[wave equation]] for the velocity potential {{mvar|φ}}. Again the oscillatory part of the velocity vector {{math|'''v'''}} is related to the velocity potential by {{math|'''v''' {{=}} ∇''φ''}}, while as before {{math|Δ}} is the [[Laplace operator]], and {{mvar|c}} is the average speed of sound in the [[transmission medium|homogeneous medium]]. Note that also the oscillatory parts of the [[pressure]] {{mvar|p}} and [[density]] {{mvar|ρ}} each individually satisfy the wave equation, in this approximation.