Notes on the Finite Element Method 6 Steady Viscous flow 8 The Time-dependent Vorticity Equation: Kelvin-Helmholtz Instability

Chapter 7 Non-linear Advection

7.1 Equations

The governing equation is now:

\bm{u}\cdot\bm{\nabla}\,\bm{u}+\bm{\nabla}p=\frac{1}{\mbox{Re}}\nabla^{2}\bm{u}.

(7.1)

along with the mass conservation equation 6.2. Here Re represents the Reynolds number, and the dimensional pressure has been nondimensionalized by $\rho\,U^{*2}$ .

7.2 Finite Element Method

7.2.1 Weak Form

Multiplied by a test function and integrated over the domain. the $x$ -component of equation 7.1 is:

\int\!\phi_{m}\left(u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y% }\right)da+\int\!\phi_{m}\frac{\partial p}{\partial x}\,da+\frac{1}{\mbox{Re}}% \int\!\left(\frac{\partial\phi_{m}}{\partial x}\frac{\partial u}{\partial x}+% \frac{\partial\phi_{m}}{\partial y}\frac{\partial u}{\partial y}\right)da=0

(7.2)

Recall the expansion for $u$ (equation 6.8, the first term on the left of 7.2is:

\sum_{n}\sum_{o}\left[\int\!\phi_{m}\left(\phi_{o}\frac{\partial\phi_{n}}{% \partial x}u_{o}+\phi_{o}\frac{\partial\phi_{n}}{\partial y}v_{o}\right)\right% ]u_{n}da

(7.3)

for each element ( $k$ ), define advection element matrices as:

Je^{x}_{k,m,n,o}=\int_{A_{k}}\!\phi_{m}\phi_{o}\frac{\partial\phi_{n}}{% \partial x}\,da_{k}\quad Je^{y}_{k,m,n,o}=\int_{A_{k}}\!\phi_{m}\phi_{o}\frac{% \partial\phi_{n}}{\partial y}\,da_{k}

(7.4)

These matrices can be evaluated initially, and reused throughout the calculation. For an element with baseline oriented an angle $\theta$ relative to the $x$ -axis: The corresponding $Je^{\eta}_{k,m,n,o}$ is:

Je^{\eta}_{k,m,n,o}=\int_{A_{k}}\!\sum_{i}\sum_{j}\sum_{l}C_{i,m}C_{j,n}C_{l,o% }q(j)I(p(i)+p(j)+p(l),q(i)+q(j)+q(l)-1)\,da_{k}

(7.5)

If the base of an element is not parallel to the $x$ -axis (as illustrated in figure LABEL:Fi:C4S12triangles, form the matrix

\mathcal{R}=\begin{bmatrix}\cos{\theta}&-\sin{\theta}\\ \sin{\theta}&\cos{\theta}\end{bmatrix}

(7.6)

Then

\begin{bmatrix}Je^{x}\\ Je^{y}\end{bmatrix}=\mathcal{R}\begin{bmatrix}Je^{\xi}\\ Je^{\eta}\end{bmatrix}

(7.7)

7.2.2 Assembly

Assembling global matrices such as mass and stiffness proceeds in the usual fashion, adding contributions from each element cookie-cutter style. The global advection matrix $\mathcal{J}$ has elements

J_{m,n}=\sum_{o}\left(\!Je^{x}_{m,n,o}u_{o}+\!Je^{y}_{m,n,o}v_{o}\right)

(7.8)

This matric is also assembled element by element, and has to be re-evaluated each time the velocity is re-evaluated.

Following chapter 6 the global problem can now be written

\begin{bmatrix}\mathcal{J}+\mathcal{K}/\mbox{Re}&0&\mathcal{G}^{x}&0\\ 0&\mathcal{J}+\mathcal{K}/\mbox{Re}&\mathcal{G}^{y}&0\\ \mathcal{D}^{x}&\mathcal{D}^{y}&0&1\\ 0&0&1&0\end{bmatrix}\begin{bmatrix}u\\ v\\ p\\ \lambda\end{bmatrix}=\mathcal{L}\bm{s}=0,\quad\mbox{where}\quad\bm{s}=\begin{% bmatrix}u\\ v\\ p\\ \lambda\end{bmatrix}

(7.9)

The Dirichlet boundary conditions are handled as in the previous chapter:

\widetilde{\mathcal{L}}\widetilde{\bm{s}}=-\widehat{\mathcal{L}}\widehat{\bm{s}}

(7.10)

The new issue is that since $\mathcal{J}$ is a priori unknown, an iterative method is needed. For these solution, the $\mbox{Re}=0$ solution is computed first, then $\mathcal{J}$ is evaluated based on the viscous solution. The process is repeated until convergence.

7.3 Results:

7.3.1 Blasius Boundary Layer

7.3.2 Flow Around a Cylinder

Viscous flow around a cylinder. The u-velocity is set to
one along the N, W and S boundaries. The solution is found in a
domain that extends from — Figure 7.1: Viscous flow around a cylinder. The u-velocity is set to one along the N, W and S boundaries. The solution is found in a domain that extends from $-3\leq x\leq 3$ and $-2\leq y\leq 2$ . Only the central portion is illustrated.