Thursday, April 30, 2015

Temporal and Spatial Stability Analysis of the Orr-Sommerfeld Equation

This is the second and final part of the stability analysis of the Orr-Sommerfeld Equation. In my previous post, I went through the derivation and nondimensionalization of the Orr-Sommerfeld Equation. In this part, I show how to perform the temporal and spatial stability analyses.

Blasius Velocity Profile

A 2-D Blasius boundary layer can be expressed as
$$\begin{align}\eta=y\sqrt{\frac{U_{\infty}}{2\nu x}}\\f'''+ff''&=0\\f(0)=f'(0)&=0\\f'(\infty)&=1\end{align}$$
This is a nonlinear ordinary differential equation that is most easily solved using a shooting method. In a shooting method, the boundary value problem is made into an initial value problem, whereby an initial guess of one parameter is iteratively updated until the opposite boundary conditions are met. In the present case, the unknown boundary condition is \(f''(0)\) , so it is iteratively adjusted until \(|f'(10)-1|\leq10^{-6}\). The present case uses a fourth-order Runge-Kutta method of integration. The initial condition The transformation of the Blasius velocity profile from \(\eta\) to \(\xi\) is performed by determining the distance from the wall, \(\delta\), where the stream-wise velocity is \(99.9\%\) of the free-stream velocity.
The figure below shows the Blasius velocity profile and its derivatives.

Temporal Stability

The Orr-Sommerfeld equation can be analyzed for temporal stability by assuming that \(\bar{\alpha}\) is real. This enables us to rearrange the non-dimensional Orr-Sommerfeld equation as follows.
If we discretize this equation using a finite difference approximation for \(\bar{\phi}''\) and \(\bar{\phi}''''\), we get an equation of the form
where \(\Phi\) is a vector containing values \(\bar{\phi}_{i}\) at discrete locations. This is nothing more than an eigenvalue problem, where \(\tilde{c}\) is a diagonal matrix of eigenvalues for the discrete system. The present calculations use a second order central differencing scheme to approximate the second and fourth derivative terms.
At the boundaries, \(\bar{\phi}=0\), so the first and last columns of \(A\) and \(B\) are removed. However, \(\bar{\phi}'=0\) also at the boundaries. Fortunately, using a backward difference method at the boundary gives \(\bar{\phi}_{-1}\approx\bar{\phi}_{0}\), so the \(\bar{\phi}_{i-2}\) term can be omitted from the \(\bar{\phi}''''\) approximation just inside the wall boundary. Similarly, the \(\bar{\phi}_{i+2}\) term can be omitted just inside the free-stream boundary.
The finite difference approximations in Equation \ref{eqfindiff} gives a pentadiagonal \(A\) and tridiagonal \(B\).
$$\begin{align}A&=\frac{1}{h^{4}}\left[\begin{array}{ccccc}a_{11} & a_{21} & a_{3} &  & 0\\a_{21} & \ddots & \ddots & \ddots\\a_{3} & \ddots & \ddots & \ddots & a_{3}\\ & \ddots & \ddots & \ddots & a_{2n}\\0 &  & a_{3} & a_{2n} & a_{1n}\end{array}\right]\\B&=\frac{1}{h^{2}}\left[\begin{array}{cccc}b_{1} & b_{2} &  & 0\\b_{2} & \ddots & \ddots\\ & \ddots & \ddots & b_{2}\\0 &  & b_{2} & b_{1}\end{array}\right]\end{align}\tag{6}\label{fdmatrix}$$
The coefficients in \(A\) depend on the velocity profile, which varies with the distance from the wall. However, \(B\) is constant for a given \(\bar{\alpha}\). In addition, \(B\) is guaranteed to be real and symmetric, and is thus guaranteed to be invertible. We can thus left multiply Equation \ref{eigmatrixform} by \(B^{-1}\) and rewrite the equation as follows.
It is clear that the eigenvalues of \(B^{-1}A\) are the eigenvalues in \(\tilde{c}\). Matlab's eig function was used to solve for the eigenvalues for each combination of \(\bar{\alpha}\) and \(Re\). The following figure shows the eigenvalues of the discrete system using one value of \(\bar{\alpha}\) and \(Re\). Each combination of \(\bar{\alpha}\) and \(Re\) gives a unique set of eigenvalues similar to these.

Because this is a stability analysis, we only care about the most unstable eigenvalue. The governing equation is related to \(e^{i\alpha(x-\left(c_{r}+ic_{i}\right)t)}\). If \(\alpha\) is real, then any instabilities must come from an eigenvalue with a positive imaginary component. For this reason, the interesting eigenvalue is the eigenvalue with the maximum imaginary component. This analysis was done over a range of \(\bar{\alpha}\) and \(Re\), and the most unstable eigenvalue was stored for each combination. Matlab's contour function was then used to plot the contours shown in the following figure.

It is important to note that the contours shown here differ from those found by Wazzan in [7] because the Orr-Sommerfeld equation has been nondimensionalized in a different manner. The present case used the boundary layer thickness \(\delta\), while Wazzan used displacement thickness \(\delta^{*}\).

Spatial Stability

The temporal stability analysis is performed assuming that \(\bar{\alpha}\) is real. In the spatial stability analysis, we assume that \(\omega\) is real. It is possible to derive a polynomial eigenvalue problem for the eigenvalues \(\bar{\alpha}\), but solving the polynomial eigenvalue problem presented some challenges. Namely, at least one of the coefficient matrices was singular.

To circumvent this challenge, a mapping method has been used to give a relation between a complex \(\bar{\alpha}\) and a complex \(\omega\). In the present case, we hold \(\bar{\alpha}_{i}\) at a specific value and vary \(\bar{\alpha}_{r}\). For each \(\bar{\alpha}_{r}\), the discrete eigenvalue problem is solved for \(\omega\), and the most unstable eigenvalue is kept as before. By varying \(\bar{\alpha}_{r}\) with \(\bar{\alpha}_{i}\), a curve of \(\omega\) values can be plotted in the complex plane, as shown in the figure above. We then determine if there is an \(\bar{\alpha}_{r}\) that gives \(\omega_{i}=0\) and store any locations that satisfy this condition. This process is repeated for a range of \(Re\) for all interesting \(\bar{\alpha}_{i}\) values. The results of this process are shown in the figure below.


This report is not the first on this topic, and the results do match fairly well with the references. Once the discretization was properly arranged, the eigenvalue computation was trivial using Matlab's built-in functions. Some factors added complexity to the project. The temporal stability curves are sensitive to the discretization step size and the number of \(Re\) and \(\bar{\alpha}\) values used. However, the spatial stability curves were far more sensitive. 

It turns out that an insufficient number of \(\bar{\alpha}_{r}\) values will give bad resolution in the complex \(\omega\) plane, periodically overestimating the zero locations. This caused some oscillations at larger values of \(Re\). Additionally, a large number of Reynold's Number steps were required to ensure that the each of the desired contours were visible.Compounding the issue was the fact that an extra variable effectively raised computation time by nearly an order of magnitude.


  1. J. D. Anderson. Fundamentals of Aerodynamics. Aeronautical and Aerospace Engineering Series. McGraw Hill, 4th edition, 2007.
  2. M. J. Maghrebi. Orr Sommerfeld Solver using Mapped Finite Di fference scheme for plane wake fl ow. Journal of Aerospace Science and Technology, 2(4):55-63, December 2005
  3. P. J. J. Moeleker. Linear Temporal Stability Analysis. Technical report, Delft University of Technology, 1998. Series 01: Aerodynamics 07.
  4. B. S. Ng and W. H. Reid. Simple Asymptotics for the Temporal Spectrum of an Orr-Sommerfeld Problem. Applied Mathematics Letters, 13:51-55, 2000.
  5. S. A. Orszag. Accurate Solution of the Orr-Sommerfeld Stability Equation. Technical report, Massachusetts Institute of Technology, May 1971.
  6. T. Patel. Stability Properties of Self-propelled Wakes. Master's Thesis, University of California, San Diego, 2012.
  7. A. R. Wazzan, T. T. Okamura, and A. M. O. Smith. Spatial and Temporal Stability Charts for the Falkner-Skan Boundary-layer Profiles. Technical report, Douglas Aircraft Company, 1968.
  8. F. M. White. Viscous Fluid Flow. McGraw-Hill Series in Mechanical Engineering. McGraw Hill, 2nd edition, 1991.

Matlab Codes

These are the Matlab files needed to perform this analysis. Please feel free to use them as you will, but please give me credit. A link to my blog is always appreciated.
Main Stability Analysis Code: cs_orr_sommerfeld_stability.m
Fourth Order Runge-Kutta Routine: cs_rk4.m

Friday, April 3, 2015

Derivation and Nondimensionalization of the Orr-Sommerfeld Equation

The Orr-Sommerfeld equation is a famous equation that can give some insight into the stability of the velocity profile of a fluid flow. This is one of two parts on the derivation and stability analysis of the Orr-Sommerfeld equation. In this post, I derive the Orr-Sommerfeld equation starting from the 2-D Navier-Stokes equations. I then show how it can be nondimensionalized. It may look like a lot of math at first glance, but it is all relatively simple.


The 2-D Navier-Stokes equations are given as follows:
\rho\frac{D\vec{V}}{Dt}&=-\nabla p+\mu\nabla^{2}\vec{V}
Letting \(V_{x}=U+u'\), \(V_{y}=V+v'\), and \(p=P+p'\) and performing a small disturbance analysis gives the small perturbation version of the Navier-Stokes Equations
$$\begin{align}\frac{\partial u'}{\partial x}+\frac{\partial v'}{\partial y}&=0\\
\frac{\partial u'}{\partial t}+U\frac{\partial u'}{\partial x}+V\frac{\partial u'}{\partial y}+u'\frac{\partial U}{\partial x}+v'\frac{\partial U}{\partial y}&=-\frac{1}{\rho}\frac{\partial p'}{\partial x}+\frac{\mu}{\rho}\left(\frac{\partial^{2}u'}{\partial x^{2}}+\frac{\partial^{2}u'}{\partial y^{2}}\right)\\
\frac{\partial v'}{\partial t}+U\frac{\partial v'}{\partial x}+V\frac{\partial v'}{\partial y}+u'\frac{\partial V}{\partial x}+v'\frac{\partial V}{\partial y}&=-\frac{1}{\rho}\frac{\partial p'}{\partial y}+\frac{\mu}{\rho}\left(\frac{\partial^{2}v'}{\partial x^{2}}+\frac{\partial^{2}v'}{\partial y^{2}}\right)
Assuming parallel flow, where \(U\approx U(y)\) and \(V\approx0\), we can simplify this to the the following form of the Navier-Stokes equations.
$$\begin{align}\frac{\partial u'}{\partial x}+\frac{\partial v'}{\partial y}&=0\\\frac{\partial u'}{\partial t}+U\frac{\partial u'}{\partial x}+v'\frac{\partial U}{\partial y}&=-\frac{1}{\rho}\frac{\partial p'}{\partial x}+\frac{\mu}{\rho}\left(\frac{\partial^{2}u'}{\partial x^{2}}+\frac{\partial^{2}u'}{\partial y^{2}}\right)\\\frac{\partial v'}{\partial t}+U\frac{\partial v'}{\partial x}&=-\frac{1}{\rho}\frac{\partial p'}{\partial y}+\frac{\mu}{\rho}\left(\frac{\partial^{2}v'}{\partial x^{2}}+\frac{\partial^{2}v'}{\partial y^{2}}\right)\end{align}\label{simp_NS}\tag{2}$$
In this analysis, disturbances are assumed to be Tollmien-Schlichting waves, with the general form as follows.
$$\begin{align}\psi&=\phi(y)e^{i(\alpha x-\omega t)}\\u'&=\frac{\partial\psi}{\partial y}=\frac{\partial\phi}{\partial y}e^{i(\alpha x-\omega t)}\\v'&=-\frac{\partial\psi}{\partial x}=-i\alpha\phi e^{i(\alpha x-\omega t)}\end{align}\tag{3}$$
The temporal and spatial derivatives are then calculated as follows.
$$\begin{align}\frac{\partial u'}{\partial t}&=-i\omega\frac{\partial\phi}{\partial y}e^{i(\alpha x-\omega t)}\\\frac{\partial u'}{\partial x}&=i\alpha\frac{\partial\phi}{\partial y}e^{i(\alpha x-\omega t)}\\\frac{\partial^{2}u'}{\partial x^{2}}&=-\alpha^{2}\frac{\partial\phi}{\partial y}e^{i(\alpha x-\omega t)}\\\frac{\partial u'}{\partial y}&=\frac{\partial^{2}\phi}{\partial y^{2}}e^{i(\alpha x-\omega t)}\\\frac{\partial^{2}u'}{\partial y^{2}}&=\frac{\partial^{3}\phi}{\partial y^{3}}e^{i(\alpha x-\omega t)}\\\frac{\partial v'}{\partial t}&=-\alpha\omega\phi e^{i(\alpha x-\omega t)}\\\frac{\partial v'}{\partial x}&=\alpha^{2}\phi e^{i(\alpha x-\omega t)}\\\frac{\partial^{2}v'}{\partial x^{2}}&=i\alpha^{3}\phi e^{i(\alpha x-\omega t)}\\\frac{\partial v'}{\partial y}&=-i\alpha\frac{\partial\phi}{\partial y}e^{i(\alpha x-\omega t)}\\\frac{\partial^{2}v'}{\partial y^{2}}&=-i\alpha\frac{\partial^{2}\phi}{\partial y^{2}}e^{i(\alpha x-\omega t)}\end{align}\tag{4}$$
We can then substitute each of these derivatives into Equation \(\ref{simp_NS}\) and we get the following relations.
$$\begin{align}e^{i(\alpha x-\omega t)}\left[i\alpha\frac{\partial\phi}{\partial y}-i\alpha\frac{\partial\phi}{\partial y}\right]&=0\\-\rho e^{i(\alpha x-\omega t)}\left[-i\omega\frac{\partial\phi}{\partial y}+i\alpha U\frac{\partial\phi}{\partial y}+-i\alpha\phi\frac{\partial U}{\partial y}-\frac{\mu}{\rho}\left(-\alpha^{2}\frac{\partial\phi}{\partial y}+\frac{\partial^{3}\phi}{\partial y^{3}}\right)\right]&=\frac{\partial p'}{\partial x}\\-\rho e^{i(\alpha x-\omega t)}\left[-\alpha\omega\phi+U\alpha^{2}\phi-\frac{\mu}{\rho}\left(i\alpha^{3}\phi-i\alpha\frac{\partial^{2}\phi}{\partial y^{2}}\right)\right]&=\frac{\partial p'}{\partial y}\end{align}$$
To eliminate the pressure fluctuation term, differentiate the x- and y-momentum equations by \(y\) and \(x\), respectively.
$$\begin{align}\frac{1}{-\rho e^{i(\alpha x-\omega t)}}\frac{\partial^{2}p'}{\partial x\partial y}&=-i\omega\frac{\partial^{2}\phi}{\partial y^{2}}+i\alpha\frac{\partial U}{\partial y}\frac{\partial\phi}{\partial y}+i\alpha U\frac{\partial^{2}\phi}{\partial y^{2}}-i\alpha\frac{\partial\phi}{\partial y}\frac{\partial U}{\partial y}\\&\quad-i\alpha\phi\frac{\partial^{2}U}{\partial y^{2}}+\frac{\mu}{\rho}\left(\alpha^{2}\frac{\partial^{2}\phi}{\partial y^{2}}-\frac{\partial^{4}\phi}{\partial y^{4}}\right)\\\frac{1}{-i\alpha\rho e^{i(\alpha x-\omega t)}}\frac{\partial^{2}p'}{\partial x\partial y}&=-\alpha\omega\phi+U\alpha^{2}\phi+\frac{\mu}{\rho}\left(-i\alpha^{3}\phi+i\alpha\frac{\partial^{2}\phi}{\partial y^{2}}\right)\end{align}\tag{5}$$
Equating the two momentum equations gives
$$-i\omega\frac{\partial^{2}\phi}{\partial y^{2}}+i\alpha U\frac{\partial^{2}\phi}{\partial y^{2}}-i\alpha\phi\frac{\partial^{2}U}{\partial y^{2}}+\frac{\mu}{\rho}\left(2\alpha^{2}\frac{\partial^{2}\phi}{\partial y^{2}}-\frac{\partial^{4}\phi}{\partial y^{4}}-\alpha^{4}\phi\right)+i\alpha^{2}\omega\phi-iU\alpha^{3}\phi=0$$
This simplifies to the Orr-Sommerfeld Equation.
$$\left(U-\frac{\omega}{\alpha}\right)\left(\frac{\partial^{2}\phi}{\partial y^{2}}-\alpha^{2}\phi\right)-\phi\frac{\partial^{2}U}{\partial y^{2}}+\frac{i\nu}{\alpha}\left(\frac{\partial^{4}\phi}{\partial y^{4}}-2\alpha^{2}\frac{\partial^{2}\phi}{\partial y^{2}}+\alpha^{4}\phi\right)=0\label{orrsommerfeld}\tag{6}$$


The Orr-Sommerfeld equation is nondimensionalized using the following nondimensional parameters,
$$\bar{U}=\frac{U}{U_{\infty}}\quad\xi=\frac{y}{\delta}\quad\bar{\phi}=\frac{\phi}{U_{\infty}\delta}\quad\bar{c}=\frac{c}{U_{\infty}}\quad\bar{\alpha}=\alpha\delta\quad Re_{\delta}=\frac{U_{\infty}\delta}{\nu}\tag{7}$$
where \(c=\frac{\omega}{\alpha}\) and \(\delta\) is the boundary layer thickness. Substituting these into Equation \(\ref{orrsommerfeld}\) gives
Applying the chain rule for each partial derivative gives
Finally, factor out \(\frac{U_{\infty}^{2}}{\delta}\) and substitute for \(Re_{\delta}\) to get
For convenience, derivatives with respect to the station coordinate \(\xi\) are hereafter denoted with prime notation. This gives the final nondimensional form of the Orr-Sommerfeld equation:

Thursday, February 5, 2015

Delaunay Triangulation

Generation of unstructured grids occurs frequently when solving numerical problems in engineering. While structured grids typically use quadrilaterals (and hexahedra in 3D) which are very simple to generate, unstructured grids use triangles (and tetrahedra in 3D), which are much more difficult to generate. Fortunately, there are many programs available to help generate these triangulations.

It is important to know how these triangulations are generated. While not the only triangulation method, one method that is almost universally included is Delaunay triangulation.  In this post, I explain what Delaunay triangulation is, why it is so popular, and the process that is followed by many programs.

What is a Delaunay triangulation?

By definition, a Delaunay triangulation is a triangulation in which no vertex lies within the circumcircle of any triangle. The circumcircle of a triangle is simply the circle that passes through the three vertices of that triangle.  Here is one very simple example.  Each domain in the image below contains four points. There are exactly two triangulations for this domain. On the left, we see that one corner of each triangle is inside the circumcircle of the other. On the right, neither circumcircle encompasses the other triangle's opposing vertex. The case on the right is a Delaunay triangulation.

Here you see the basic concept behind an implementation. First generate a triangulation, then check to see if any vertices fall within the cirumcircle of another triangle. If they do, simply swap the diagonal separating the two triangles. This will always give a Delaunay triangulation.

Why is it so popular?

Delaunay triangulation is fairly simple conceptually, but why is it so popular? The primary reason for its popularity is that the resulting mesh is inherently good quality.  For a two-dimensional Delaunay triangulation, it can be shown that the minimum interior angle of each triangle is maximized, and that the maximum interior angle is minimized. The resulting triangles are as equiangular as possible. Another benefit is that the triangulation is independent of the order in which nodes are placed. It is also a relatively simple triangulation method to implement for convex hulls (think of a convex hull as a domain with no "dents" on the boundary). Implementation for non-convex hulls are a bit more challenging, but not impossible.

The Process

The algorithm presented here is as described by S. W. Sloan in Ref. [1]. This routine generates a Delaunay triangulation for a set of predetermined coordinates.

First, generate a triangle that encloses the entire domain to be triangulated.  This triangle is called the supertriangle. The size and shape of the supertriangle is irrelevant, as long as all points in the domain are contained within.

The next portion of the algorithm is an iterative process. For each point in the domain, add the point to the triangulation by subdividing the triangle containing that point. You will then have a new node surrounded by three triangles, seen in green in the image below, and across the far edge of each new triangle is an opposing node, shown in red.

For each new triangle created, if the opposing node for that triangle is not part of the supertriangle, check whether the opposing node is within the circumcircle of the new triangle. If it is, swap the diagonal separating the newly inserted node and the opposing node. In the example, two diagonals need to be swapped.

Swapping the diagonal between two triangles gives two new triangles. Again, a circumcircle test is required for the new opposing nodes, and the process continues. until the circumcircle test is satisfied for all triangles connected to the recently inserted node.

The entire process then repeats until all nodes in the domain have been inserted and triangulated.  The final step is to remove any triangles connected to the supertriangle vertices, leaving only the domain of interest. Below is a simple example to demonstrate the process one step at a time.

Extension into Three Dimensions

It is not difficult to see that Delaunay triangulation is possible in three dimensions.  The obvious difference for a tetrahedral mesh is that a circumsphere is used instead of a circumcircle. One difference that is not so obvious is that instead of swapping the edge dividing two triangles, you need to swap a dividing face. However, when swapping a face, there are two alternative positions, so some care needs to be taken when choosing which option to use.

Potential Uses

Delaunay triangulation, or any triangulation scheme for that matter, is great for connecting a known set of data points.  I have used this in conjunction with barycentric interpolation to create a program that quickly interpolates to find values between known data points. It can also be used to generate a mesh for finite element and finite volume programs. Because the nodes can be inserted in an arbitrary order, it could even be used as a strategy to adapt a mesh in real-time.

If this post was helpful, let me know in the comments. Ask questions if I have left something unclear, and I'll try to elaborate on the subject.


  1. S. W. Sloan, "A fast algorithm for constructing Delaunay triangulations in the plane", Adv. Eng. Software, Vol. 9, No. 1, pp. 34-55 1987