Potential flow

s and observing the result. for the incompressible potential flow around a circular cylinder in a uniform onflow. In potential or irrotational flow, the vorticity vector field is zero, i.e., \boldsymbol\omega \equiv \nabla\times\mathbf v=0, where \mathbf v(\mathbf x,t) is the velocity field and \boldsymbol\omega(\mathbf x,t) is the vorticity field. Like any vector field having zero curl, the velocity field can be expressed as the gradient of certain scalar, say \varphi(\mathbf x,t) which is called the velocity potential, since the curl of the gradient is always zero. We therefore have \mathbf{v} = \nabla \varphi. The velocity potential is not uniquely defined since one can add to it an arbitrary function of time, say f(t), without affecting the relevant physical quantity which is \mathbf v. The non-uniqueness is usually removed by suitably selecting appropriate initial or boundary conditions satisfied by \varphi and as such the procedure may vary from one problem to another. In potential flow, the circulation \Gamma around any simply-connected contour C is zero. This can be shown using the Stokes theorem, \Gamma \equiv \oint_C \mathbf v\cdot d\mathbf l = \int \boldsymbol\omega\cdot d\mathbf f=0 where d\mathbf l is the line element on the contour and d\mathbf f is the area element of any surface bounded by the contour. In multiply-connected space (say, around a contour enclosing solid body in two dimensions or around a contour enclosing a torus in three-dimensions) or in the presence of concentrated vortices, (say, in the so-called irrotational vortices or point vortices, or in smoke rings), the circulation \Gamma need not be zero. In the former case, Stokes theorem cannot be applied and in the later case, \boldsymbol\omega is non-zero within the region bounded by the contour. Around a contour encircling an infinitely long solid cylinder with which the contour loops N times, we have \Gamma = N \kappa where \kappa is a cyclic constant. This example belongs to a doubly-connected space. In an n-tuply connected space, there are n-1 such cyclic constants, namely, \kappa_1,\kappa_2,\dots,\kappa_{n-1}. ==Incompressible flow==

In case of an incompressible flow — for instance of a liquid, or a gas at low Mach numbers; but not for sound waves — the velocity has zero divergence: \nabla \cdot \mathbf{v} =0 \,, Substituting here \mathbf v = \nabla\varphi shows that \varphi satisfies the Laplace equation \nabla^2 \varphi = 0 \,, where is the Laplace operator (sometimes also written ). Since solutions of the Laplace equation are harmonic functions, every harmonic function represents a potential flow solution. As evident, in the incompressible case, the velocity field is determined completely from its kinematics: the assumptions of irrotationality and zero divergence of flow. Dynamics in connection with the momentum equations, only have to be applied afterwards, if one is interested in computing pressure field: for instance for flow around airfoils through the use of Bernoulli's principle. In incompressible flows, contrary to common misconception, the potential flow indeed satisfies the full Navier–Stokes equations, not just the Euler equations, because the viscous term \mu\nabla^2\mathbf v = \mu\nabla(\nabla\cdot\mathbf v)-\mu\nabla\times\boldsymbol\omega=0 is identically zero. It is the inability of the potential flow to satisfy the required boundary conditions, especially near solid boundaries, makes it invalid in representing the required flow field. If the potential flow satisfies the necessary conditions, then it is the required solution of the incompressible Navier–Stokes equations. In two dimensions, with the help of the harmonic function \varphi and its conjugate harmonic function \psi (stream function), incompressible potential flow reduces to a very simple system that is analyzed using complex analysis (see below). ==Compressible flow==

Steady flow Potential flow theory can also be used to model irrotational compressible flow. The derivation of the governing equation for \varphi from Eulers equation is quite straightforward. The continuity and the (potential flow) momentum equations for steady flows are given by \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 where the last equation follows from the fact that entropy is constant for a fluid particle and that square of the sound speed is c^2=(\partial p/\partial\rho)_s. Eliminating \nabla\rho from the two governing equations results in c^2\nabla\cdot\mathbf v - \mathbf v\cdot (\mathbf v \cdot \nabla)\mathbf v=0. The incompressible version emerges in the limit c\to\infty. Substituting here \mathbf v=\nabla\varphi results in (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 where c=c(v) is expressed as a function of the velocity magnitude v^2=(\nabla\phi)^2. For a polytropic gas, c^2 = (\gamma-1)(h_0-v^2/2), where \gamma is the specific heat ratio and h_0 is the stagnation enthalpy. In two dimensions, the equation simplifies to (c^2-\varphi_x^2)\varphi_{xx}+(c^2-\varphi_y^2)\varphi_{yy}-2\varphi_x\varphi_y\varphi_{xy}=0. 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 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 \nabla (h+v^2/2) - \mathbf v\times\boldsymbol\omega = T \nabla s where h is the specific enthalpy, \boldsymbol\omega is the vorticity field, T is the temperature and s is the specific entropy. Since in front of the leading shock wave, we have a potential flow, Bernoulli's equation shows that h+v^2/2 is constant, which is also constant across the shock wave (Rankine–Hugoniot conditions) and therefore we can write \mathbf v\times\boldsymbol\omega = -T \nabla s 1) When the shock wave is of constant intensity, the entropy discontinuity across the shock wave is also constant i.e., \nabla s=0 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 \nabla s 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 U\mathbf{e}_x be the mainstream and consider small deviations from this velocity field. The corresponding velocity potential can be written as \varphi = x U + \phi where \phi characterizes the small departure from the uniform flow and satisfies the linearized version of the full equation. This is given by (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 where M=U/c_\infty is the constant Mach number corresponding to the uniform flow. This equation is valid provided M is not close to unity. When |M-1| is small (transonic flow), we have the following nonlinear equation 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} where \alpha_* is the critical value of Landau derivative \alpha = (c^4/2\upsilon^3)(\partial^2 \upsilon/\partial p^2)_s and \upsilon=1/\rho is the specific volume. The transonic flow is completely characterized by the single parameter \alpha_*, which for polytropic gas takes the value \alpha_*=\alpha=(\gamma+1)/2. Under hodograph transformation, the transonic equation in two-dimensions becomes the Euler–Tricomi equation. Unsteady flow The continuity and the (potential flow) momentum equations for unsteady flows are given by \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. The first integral of the (potential flow) momentum equation is given by \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} where f(t) is an arbitrary function. Without loss of generality, we can set f(t)=0 since \varphi is not uniquely defined. Combining these equations, we obtain \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. Substituting here \mathbf v=\nabla\varphi results in \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}). Nearly parallel flows: As in before, for nearly parallel flows, we can write (after introducing a recaled time \tau=c_\infty t) \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} provided the constant Mach number M is not close to unity. When |M-1| is small (transonic flow), we have the following nonlinear equation \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}. Sound waves: In sound waves, the velocity magnitude v (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, c is a constant (for example, in polytropic gas c^2=(\gamma-1)h_0), we have \frac{\partial^2 \varphi}{\partial t^2} = c^2 \nabla^2 \varphi, which is a linear wave equation for the velocity potential . Again the oscillatory part of the velocity vector is related to the velocity potential by , while as before is the Laplace operator, and is the average speed of sound in the homogeneous medium. Note that also the oscillatory parts of the pressure and density each individually satisfy the wave equation, in this approximation. ==Applicability and limitations==

Potential flow does not include all the characteristics of flows that are encountered in the real world. Potential flow theory cannot be applied for viscous internal flows, Incompressible potential flow also makes a number of invalid predictions, such as d'Alembert's paradox, which states that the drag on any object moving through an infinite fluid otherwise at rest is zero. More precisely, potential flow cannot account for the behaviour of flows that include a boundary layer. The basic idea is to use a holomorphic (also called analytic) or meromorphic function , which maps the physical domain to the transformed domain . While , , and are all real valued, it is convenient to define the complex quantities \begin{align} z &= x + iy \,, \text{ and } & w &= \varphi + i\psi \,. \end{align} Now, if we write the mapping as \begin{align} f(x + iy) &= \varphi + i\psi \,, \text{ or } & f(z) &= w \,. \end{align} Then, because is a holomorphic or meromorphic function, it has to satisfy the Cauchy–Riemann equations \begin{align} \frac{\partial\varphi}{\partial x} &= \frac{\partial\psi}{\partial y} \,, & \frac{\partial\varphi}{\partial y} &= -\frac{\partial\psi}{\partial x} \,. \end{align} The velocity components , in the directions respectively, can be obtained directly from by differentiating with respect to . That is \frac{df}{dz} = u - iv So the velocity field is specified by \begin{align} u &= \frac{\partial\varphi}{\partial x} = \frac{\partial\psi}{\partial y}, & v &= \frac{\partial\varphi}{\partial y} = -\frac{\partial\psi}{\partial x} \,. \end{align} Both and then satisfy Laplace's equation: \begin{align} \Delta\varphi &= \frac{\partial^2\varphi}{\partial x^2} + \frac{\partial^2\varphi}{\partial y^2} = 0 \,,\text{ and } & \Delta\psi &= \frac{\partial^2\psi}{\partial x^2} + \frac{\partial^2\psi}{\partial y^2} = 0 \,. \end{align} So can be identified as the velocity potential and is called the stream function. Lines of constant are known as streamlines and lines of constant are known as equipotential lines (see equipotential surface). Streamlines and equipotential lines are orthogonal to each other, since \nabla \varphi \cdot \nabla \psi = \frac{\partial\varphi}{\partial x} \frac{\partial\psi}{\partial x} + \frac{\partial\varphi}{\partial y} \frac{\partial\psi}{\partial y} = \frac{\partial \psi}{\partial y} \frac{\partial \psi}{\partial x} - \frac{\partial \psi}{\partial x} \frac{\partial \psi}{\partial y} = 0 \,. Thus the flow occurs along the lines of constant and at right angles to the lines of constant . is also satisfied, this relation being equivalent to . So the flow is irrotational. The automatic condition then gives the incompressibility constraint . ==Examples of two-dimensional incompressible flows==

Any differentiable function may be used for . The examples that follow use a variety of elementary functions; special functions may also be used. Note that multi-valued functions such as the natural logarithm may be used, but attention must be confined to a single Riemann surface. Power laws In case the following power-law conformal map is applied, from to : w=Az^n \,, then, writing in polar coordinates as , we have Line source and sink A line source or sink of strength Q (Q>0 for source and Q for sink) is given by the potential w = \frac{Q}{2\pi} \ln z where Q in fact is the volume flux per unit length across a surface enclosing the source or sink. The velocity field in polar coordinates are u_r = \frac{Q}{2\pi r},\quad u_\theta=0 i.e., a purely radial flow. Line vortex A line vortex of strength \Gamma is given by w=\frac{\Gamma}{2\pi i}\ln z where \Gamma is the circulation around any simple closed contour enclosing the vortex. The velocity field in polar coordinates are u_r = 0,\quad u_\theta=\frac{\Gamma}{2\pi r} i.e., a purely azimuthal flow. ==Analysis for three-dimensional incompressible flows==

For three-dimensional flows, complex potential cannot be obtained. Point source and sink The velocity potential of a point source or sink of strength Q (Q>0 for source and Q for sink) in spherical polar coordinates is given by \phi = -\frac{Q}{4\pi r} where Q in fact is the volume flux across a closed surface enclosing the source or sink. The velocity field in spherical polar coordinates are u_r = \frac{Q}{4\pi r^2}, \quad u_\theta=0, \quad u_\phi = 0. == See also ==