Methodology

The methodology used here is based on the nonlinear disturbance equations, which is a newly developed numerical method [1]. From the general Navier-Stokes equations in a Cartesian coordinate system:

 

$$\frac{\partial q}{\partial t} + \frac{\partial F}{\partial x... ... \frac{\partial S}{\partial y} + \frac{\partial E}{\partial z}$$ (1)

The flow field is then split into a mean and a fluctuating part:

 

$${q} = q_o + q^\prime$$ (2)

where

$$q_o = \lim_{T \rightarrow \infty} \frac{1}{T} \int_{t_0}^{t_0 + T} q(t) dt$$ (3)

Substitution of equation (2) into (1) and rearranging results in the nonlinear disturbance equations(NLDE):

 

$$\frac{\partial q^\prime}{\partial t} + \frac{\partial F^\p... ...me}{\partial y} + \frac{\partial H_n^\prime}{\partial z} = {Q}$$ (4)

Where

 

$$q^\prime = \left\{ \begin{array} {c} \rho^\prime \\ \rh... ...e w_o + \rho^\prime w^\prime \\ e^\prime \end{array} \right\}$$ (5)

On the left hand side of the NLDE are terms related to the perturbation properties and the cross terms, whereas the right hand side has strictly mean flow terms. The perturbation terms contain linear and nonlinear quantities:

$$F^\prime = \left\{ \begin{array} {c} \rho_o u^\prime + \rh... ... (e_o + p_o) + u_o (e^\prime + p^\prime) \end{array} \right\}$$ (6)

$$G^\prime = \left\{ \begin{array} {c} \rho_o v^\prime + \rh... ... (e_o + p_o) + v_o (e^\prime + p^\prime) \end{array} \right\}$$ (7)

$$H^\prime = \left\{ \begin{array} {c} \rho_o w^\prime + \rh... ... (e_o + p_o) + w_o (e^\prime + p^\prime) \end{array} \right\}$$ (8)

and

$$F_n^\prime = \left\{ \begin{array} {c} \rho^\prime u^\prim... ...ime \\ u^\prime (e^\prime + p^\prime) \end{array} \right\}$$ (9)

$$G_n^\prime = \left\{ \begin{array} {c} \rho^\prime v^\prim... ...ime \\ v^\prime (e^\prime + p^\prime) \end{array} \right\}$$ (10)

$$H_n^\prime = \left\{ \begin{array} {c} \rho^\prime w^\prim... ...^2} \\ w^\prime (e^\prime + p^\prime) \end{array} \right\}$$ (11)

The mean flow source term Q is assumed to be time independent:

$${Q} = -\left( \frac{\partial F_o}{\partial x} + \frac{\pa... ...c{\partial S_o}{\partial y} + \frac{\partial E_o}{\partial z}$$ (12)

If the NLDE is averaged, it becomes the Reynolds-averaged Navier-Stokes equation, where the Reynold's stresses are on the left hand side. Thus, for a laminar flow Q=0.

We seek a solution of the perturbation variables $q^\prime$ with a known mean flow field which can be obtained from existing well-developed CFD codes (e.g. CFL3D, INS3D, OVERFLOW, ...) for steady flow. The CFD code for mean flow is a lower order method that is ineffective for unsteady flow due to their inherent dissipation and dispersion. The NLDE method is a higher order accurate method that requires much fewer grid points than a lower order method. This methodology allows us to use the most effective algorithms for the steady and unsteady portions of field, respectively. NLDE combined with characteristic type boundary conditions makes introducing various wind conditions straight-forward. It also minimizes round-off error since we are only computing perturbations. More discussion on this new method is in [1].