How to add perturbation to a base state in physical grid space?

Question

For study 3D instabilities problem i.e a base state (velocity field and temperature field) and an additional perturbation.

The method must be independent of the type of instabilities considered but as an example, let's take the well known Kelvin-Helmholtz instability. The dynamical equation for the system can be defined as the Boussinesq equation \begin{equation} \nabla\cdot\mathbf{u}=0 \end{equation} \begin{equation} \frac{d\mathbf{u}}{dt}=-\frac{1}{\rho}\nabla p + b \mathbf{n} + \nu_\mathbf{u} \Delta \mathbf{u} \end{equation} \begin{equation} \frac{d b}{dt}= \nu_{b} \Delta b \end{equation}

Where $\mathbf{u}=(u,v,w)$ is the velocity vector, $b$ is the buoyancy proportional to the temperature, $p$ is the pressure, $\frac{d~}{dt}$ is the lagrangian derivative, $\nu_\mathbf{u}$ is the kinematic viscosity, $\nu_{b}$ is the diffusion coefficient of buoyancy, $\Delta$ is the Laplacien operator and $\nabla$ is the gradient operator. The problem is define on $[0,Lx]\times[0,Ly]\times[0,Lz]$. The problem is $Lx$ periodic. $\mathbf{u}$ and $b$ are the computational variables. We initialize the field with \begin{equation} \mathbf{u}(\mathbf{x},t=0)=\mathbf{u}_B(\mathbf{x})+\mathbf{u}_P(\mathbf{x}) \end{equation} \begin{equation} b(\mathbf{x},t=0)=b_B(\mathbf{x})+b_P(\mathbf{x}) \end{equation} where $\mathbf{x}=(x,y,z)$ is the spacial coordonate, $t$ is the time coordinate, the sub-script $B$ is for the base state and the sub-script $P$ is for the perturbed field.

if $z>Lz/2$ then $u_B(\mathbf{x})=\alpha_1$ else $u_B(\mathbf{x})=\alpha_2$. if $z>Lz/2$ then $u_B(\mathbf{x})=\beta_1$ else $u_B(\mathbf{x})=\beta_2$. $v_B=0$ and $w_B=0$.

Numerically we add a perturbation $(\mathbf{u}_P,b_P)$ for accelerate the developmental of the instability. We expect the perturbed field to be Gaussian and negligible in front of the base state. By negligeable let's said $||(\mathbf{u}_P,b_P)||\approx 10^{-3}||(\mathbf{u}_B,b_B)||$. The flow is discretized on a regular structured grid.

How to implement properly the perturbed field $(\mathbf{u}_P,b_P)$ in the physical grid space ?

Answer 1

My experience in these types of problems is that almost any perturbation will do. I suggest you do not try to perturb $u$ or $p$, because generating a divergence-free $u$ with a perturbed component is more complicated than necessary. You can easily perturb $b$ with lots of different things including a small random field or a randomly-distributed set of Gaussian bubbles or small sinusoids with randomly-chosen frequencies.

The problem in my experience is that you need to choose to kick your problem with something appropriate to the physics and the boundary conditions. If the perturbation in the Kelvin-Helmholtz problem is too lopsided, you might find yourself with a very long transient while the the problem settles into a more symmetric pattern. Also, depending on the dynamics of landscape, you may land in one kind of semi-stable flow pattern rather than another.

Update: Here's some math that might be used as initial conditions on $u$ and $b$: $$ u_P(\mathbf{x})=0 \\ b_P(\mathbf{x})=\epsilon{\rm random}(\mathbf{x}) $$ where ${\rm random}(\mathbf{x})$ gives you a random number between 0 and 1 at each point in space, and $\epsilon$ is a small parameter.

The other cases proceed similarly.