Chapter 4: Advanced Potential Flow Theory

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4 Chapter 4: Advanced Potential Flow Theory

Potential flow is typically limited in terms of its application to the previous discussion. There are higher levels of the theory that are important to understand. Hence, we develop potential flow to something called the Complex Potential. We also explore the concept of extensions to compressible flows (or flows where density is changing).

Complex Potential

Complex potential flow refers to a method in fluid mechanics that models these potential flows as a complex function. Even though potential flow is already complicated, this concept introduces an imaginary number component and mapping of coordinate systems that can be confusing. This is, however, a simplified approach to potential flow that has classically helped.

Basic Concepts

Visualizing complex theory can be tough, for this reason, consider a quick review. Two good sites to visually (and simply describe) complex math can be found in this discussion or these notes. In reviewing this, recall that complex math typically utilizes the x-y plan as the real-imaginary axis. A few comments about complex mathematics prior to getting started.

The foundation is that $i=\sqrt{-1}$ . With this, we can develop complex math rules. Keep in mind complex analysis demands a bit of brain rewiring.

Complex math basically treats the y-axis as the imaginary axis (or magnitude attached to $i$ ). That is the real numbers drive the x-axis and the imaginary magnitude is the y-axis. For example: $Y=a+ib$ can be considered similar to $\left(a,b\right)$ .

In a general sense, we seek to find function in the complex number regime. So we define a complex result, $w$ , which is a function of a complex input, $z$ .

(1) $\begin{equation*} w=u+iv=f(z) \end{equation*}$

where $z=x+iy$ . It is important to think of $z$ as the input, but rather than $x$ , it is a complex number defined as $x+iy$ . Note that $\theta=tan^{-1}\left(\frac{y}{x}\right)$ .

Recall Euler’s identity:

(2) $\begin{equation*} e^{i\theta} = \cos \theta + i \sin \theta \end{equation*}$

and

(3) $\begin{equation*} e^{-i\theta} = \cos \theta - i \sin \theta \end{equation*}$

Complex conjugate (or reciprocal relationship) is defined as:

(4) $\begin{equation*} \bar{Y}=a-ib \end{equation*}$

Complex math rules for $Y=a+ib$ and $X=c+id$
Addition:

(5) $\begin{equation*} X+Y=\left(a+c\right)+i\left(b+d\right) \end{equation*}$

Multiplication:

(6) $\begin{equation*} XY=\left(a+ib\right)\left(c+id\right)=ac+iad+ibc+i^2bd= \left(ac-bd\right)+i\left(ad+bc\right) \end{equation*}$

Multiply by conjugate:

(7) $\begin{equation*} Y\barY=\left(a+ib\right)\left(a-ib\right)= aa+iab-iab+i^2bb= a^2+b^2 \end{equation*}$

The complex variables should also not be thought of as constants but, rather, functions. Hence, consider a Z-plane function with coordinates x and y.

(8) $\begin{equation*} f(z)=g\left(x,y\right)+ih\left(x,y\right) \end{equation*}$

The differential of $f(z)$ slightly differs from conventional calculus (and is described in additional detail here). A differential is still described (about a point $z_0$ ) as

(9) $\begin{equation*} f'(z_0)=\frac{dw}{dz}=\frac{\Delta w}{\Delta z} \end{equation*}$

Where $\Delta w= w\left(z_0+\Delta z\right)+-w\left(z_0\right)$ . Here the main difference is we have real and imaginary parts. This is generally handled as follows

(10) $\begin{equation*} f'(x)=\lim_{\Delta z\rightarrow 0}\frac{\Delta g\left(x,y\right)+i\Delta h\left(x,y\right)}{\Delta x+i\Delta y}. \end{equation*}$

Note here, we have the real direction (for g and x) and imaginary direction (for h and y), hence, a differential has 2 directions given by

(11) $\begin{equation*} f_x'(z_0)=\lim_{\Delta x\rightarrow 0}\frac{\Delta g\left(x,y\right)+i\Delta h\left(x,y\right)}{\Delta x}=\frac{\partial g}{\partial x}+i\frac{\partial h}{\partial x} \end{equation*}$

(12) $\begin{equation*} f_y'(z_0)=\lim_{\Delta iy\rightarrow 0}\frac{\Delta g\left(x,y\right)+i\Delta h\left(x,y\right)}{\Delta iy}=\frac{\partial g}{\partial y}+i\frac{\partial h}{i\partial y}=-i\frac{\partial g}{\partial y}+\frac{\partial h}{\partial y} \end{equation*}$

Note that $i^-1=-i$ . This is peculiar as, although a differential in $z$ is one dimension, it needs to be treated similarly to 2D and it needs to approach the same value from any direction. We can simplify the requirements as

(13) $\begin{equation*} f'(z_0)=\lim_{\Delta x \rightarrow 0}f'(z_0+\Delta x)=\lim_{i\Delta y \rightarrow 0}f'(z_0+i\Delta y), \end{equation*}$

or that the real and imaginary parts of the differentials match, i.e, respectively given as

(14) $\begin{equation*} \frac{\partial g}{\partial x}=\frac{\partial h}{\partial y} \end{equation*}$

and

(15) $\begin{equation*} \frac{\partial h}{\partial x}=-\frac{\partial g}{\partial y}. \end{equation*}$

These are known as the Cauchy-Riemann conditions. These complex concepts are widely used in classical aerodynamics.

Potential Flow Represented in Complex Theory

Now let’s take these concepts back to potential flow. Based on the definition of velocities in $\phi$ and $\psi$ , we can easily note that

(16) $\begin{equation*} u=\frac{\partial \phi}{\partial x} = \frac{\partial \psi}{\partial y} \end{equation*}$

and

(17) $\begin{equation*} v=\frac{\partial \phi}{\partial y} = -\frac{\partial \psi}{\partial x} \end{equation*}$

These interestingly are ideally matched to the aforementioned Cauchy-Riemann conditions and are a result of the kinematic requirement that $\phi$ and $\psi$ are mutually perpendicular. In the context of potential flow, the Cauchy-Riemann conditions simply imply the constraints of the definitions of u and v in the context of $\phi$ and $\psi$ .

The aforementioned discussion gives us a method to utilize complex math and leads to the definition of the complex potential function, $F$ , given as

(18) $\begin{equation*} F(z)=\phi+i\psi \end{equation*}$

In this context, note that both the real and imaginary parts satisfy Laplace’s equation.
And the complex derivatives of F yield a representation of the velocity field as:

(19) $\begin{equation*} F'=\frac{dF}{dY}=W(Y)=u-iv. \end{equation*}$

Which is referred the complex conjugate of the velocity (Eq.4) and W is the complex velocity. Such a methodology forms a framework for fluids applications with several examples given here.

The results from this model yield the theory related to the Joukowski airfoils.

Compressible Potential Flow

In a compressible potential flow, changes in fluid density are accounted for by the changes in fluid pressure and demands including the impact of sound speed. As the fluid flows, its velocity and pressure can change, causing changes in density, but the flow remains irrotational. This means that the velocity field can be represented by a scalar potential function that satisfies Laplace’s equation, which governs the behavior of the flow.
Compressible potential flow is used in various engineering and scientific applications, such as aerodynamics, gas dynamics, and combustion. It is also used to model the flow of sound waves in a fluid, which are treated as small perturbations of the fluid velocity field.

In any isentropic flow, the sound speed ( $a$ ) is defined as

(20) $\begin{equation*} a^2=\frac{\partial p}{\partial \rho}. \end{equation*}$

At a fundamental level, this implies that sound speed indirectly correlates to how density responds to pressure. That is, if a change in pressure results in a large change in density, the fluid is highly “compressible” and yields a low speed. This typically relates to air which typically has a sound speed of around 340 m/s. Alternatively, if we pressurize water, the density does not change much and leads to a higher sound speed (water is typically 1500m/s). Hence, sound speed is a result of the repsonse of density to pressure waves.

Recall the steady 2D governing equations. Conservation of mass can be written as

(21) $\begin{equation*} \frac{\partial \rho u}{\partial x}+\frac{\partial \rho v}{\partial y}=u \frac{\partial \rho}{\partial x}+\rho \frac{\partial u}{\partial x}+v\frac{\partial \rho}{\partial y}+\rho\frac{\partial v}{\partial y}=0 \end{equation*}$

which approximately simplifies to

(22) $\begin{equation*} \frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=-\frac{1}{\rho}\left(u\frac{\partial \rho}{\partial x}+v\frac{\partial \rho}{\partial y}\right). \end{equation*}$

The inviscid (weakly compressible) momentum equations are given as

(23) $\begin{equation*} u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}=-\frac{1}{\rho}\frac{\partial p}{\partial x} \end{equation*}$

(24) $\begin{equation*} u\frac{\partial v}{\partial x}+v\frac{\partial v}{\partial y}=-\frac{1}{\rho}\frac{\partial p}{\partial y} \end{equation*}$

We then sum Eqns. (23) $\times u$ and (24) $\times v$ and sub in the sound speed definition, Eq. (20), into the pressure gradient term (i.e., $\frac{\partial p}{\partial x}=a^2\frac{\partial \rho}{\partial x}$ and $\frac{\partial p}{\partial x}=a^2\frac{\partial \rho}{\partial y}$ ) to get

(25) $\begin{equation*} u^2\frac{\partial u}{\partial x}+vu\frac{\partial u}{\partial y} +uv\frac{\partial v}{\partial x}+v^2\frac{\partial v}{\partial y} =-\frac{a^2}{\rho}\left(u\frac{\partial\rho}{\partial x} + v\frac{\partial\rho}{\partial y} \right) \end{equation*}$

We then see that the RHS can utilize the conservation of mass form from Eq. (22) to obtain

(26) $\begin{equation*} u^2\frac{\partial u}{\partial x}+vu\frac{\partial u}{\partial y} +uv\frac{\partial v}{\partial x}+v^2\frac{\partial v}{\partial y} =a^2\left(\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y} \right) \end{equation*}$

Dividing by $a^2$ we obtain

(27) $\begin{equation*} \frac{u^2}{a^2}\frac{\partial u}{\partial x}+\frac{uv}{a^2}\frac{\partial u}{\partial y} +\frac{uv}{a^2}\frac{\partial v}{\partial x}+\frac{v^2}{a^2}\frac{\partial v}{\partial y} =\left(\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y} \right) \end{equation*}$

Then rearranging we obtain

(28) $\begin{equation*} \left(1-\frac{u^2}{a^2}\right)\frac{\partial u}{\partial x} +\left(1-\frac{v^2}{a^2}\right)\frac{\partial v}{\partial y} -\frac{uv}{a^2}\left(\frac{\partial u}{\partial y}+\frac{\partial v}{\partial x}\right) =0 \end{equation*}$

We then consider the irrotational aspect of potential flow, we realize that $\nabla\times\bar{V}=0$ , or that $\frac{\partial u}{\partial y}=\frac{\partial v}{\partial x}$ .

(29) $\begin{equation*} \left(1-\frac{u^2}{a^2}\right)\frac{\partial u}{\partial x} +\left(1-\frac{v^2}{a^2}\right)\frac{\partial v}{\partial y} -2\frac{uv}{a^2}\left(\frac{\partial u}{\partial y}\right) =0. \end{equation*}$

Then substituting in the definition of directional Mach number (i.e., $M_x=u/a$ and $M_y=v/a$ ), we get

(30) $\begin{equation*} \left(1-M_x^2\right)\frac{\partial u}{\partial x} +\left(1-M_y^2\right)\frac{\partial v}{\partial y} -2M_xM_y\left(\frac{\partial u}{\partial y}\right) =0. \end{equation*}$

We seek to explore the limit where $M_y<<M_x$ , i.e., flow is dominant along the x-axis, which enables the elimination of the 3rd term. We can expand this to a perturbation form from an axial free-stream velocity $U_\infty$ where

(31) $\begin{equation*} u=U_\infty+u'=U_\infty+\frac{\partial \phi}{\partial x} \end{equation*}$

and

(32) $\begin{equation*} v=v'=\frac{\partial \phi}{\partial y}. \end{equation*}$

The resulting equation becomes

(33) $\begin{equation*} \left(1-M_\infty^2\right)\frac{\partial^2 \phi}{\partial x^2} +\frac{\partial^2 \phi}{\partial y^2} =0. \end{equation*}$

We can then perform the Prandtl-Glauert transformation as

(34) $\begin{equation*} x_M=\frac{x}{\sqrt{1-M_\infty^2}} \end{equation*}$

(35) $\begin{equation*} y_M=y. \end{equation*}$

Note that $\frac{\partial}{\partial x_M}=\left(1-M_\infty^2\right)^-1\frac{\partial}{\partial x}$ . Hence, Laplace’s equation is recovered in the context of $x_M$ and $y_M$ . Finally, we obtain a slightly modified defintion for pressure coefficent ( $C_P=\frac{p-p_\infty}{0.5\rho U_\infty^2}$ ). In this transformation and in potential flow, the result yields

(36) $\begin{equation*} C_P=-\frac{2}{0.5 \rho U_\infty ^2}\frac{\partial \phi }{\partial x_M}=-\frac{2}{0.5 \rho U_\infty ^2}\frac{1}{\sqrt{1-M_\infty^2}}\frac{\partial \phi }{\partial x}. \end{equation*}$

The results from this model form of potential flow relate later to the development of transonic aerodyanmics and the Prandtl–Glauert correction.

4 Chapter 4: Advanced Potential Flow Theory

Complex Potential

Basic Concepts

Potential Flow Represented in Complex Theory

Compressible Potential Flow

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