# Presentation: 344023

- Conservation Equations

## Dynamics    

- We now consider the *Dynamics* in Computational Fluid Dynamics (CFD).
- Specifically for WIND:
- Fluid Dynamics
- Turbulence
- Chemical Dynamics
- Magneto-Fluid Dynamics
- Dynamics involves relating the forces in the system to the properties of the
- system. Dynamics involves *kinematics* (geometrical aspects of motion) and
*kinetics* (analysis of forces causing motion).
- All of these involve fluid behavior and it is known that conservation
- principles governs this behavior.

## Conservation Principles    

- The dynamics of fluid flow is described by several conservation principles:
- Mass (mass of system and species)
- Momentum
- Energy (internal, turbulent, chemical)
- Magnetism?
- Electrical charge?
- Fluid motion requires determining the fluid velocity components     and
- additional static properties (*p, T, ****, c**i*).

## Conservation Statement    

- nature of the conservation:
*Q* is conserved quantity, scalar or vector (i.e. mass, momentum)
*V* is the control volume (time-varying)
*S* is the control surface (time-varying)
-    is surface normal vector with positive direction out of volume
**D** is non-convective terms of the flux (vector or tensor)
- The control volume can translate and rotate, as well as, change shape and
- deform.  Gauss’ theorem relates control surface and control volume.

- Conservation Statement

- Time-rate of change of Q in the control volume.

- Flux of Q leaving the control volume through the control surface.

- Production of Q within the control volume.

## Equations Sets    

*Q*, **D**, and *P* can be scalars or an algebraic vectors with elements as
- vectors.  **D** can be a vector or dyadic, depending on whether Q is a scalar
- or vector.

- Equations Sets

## Complete Description of Equations    

- A complete description of the dynamic equations requires:
- Equations
- Frame of Reference
- Control Volume and Control Surface
- Boundary Conditions
- Initial Conditions

## Frames of Reference    

- Some frames of reference:
- _Inertial Cartesian Frame_
- _Cylindrical Frame_
- _Rotating Frame_
- _6 DOF Frame (generalized translation and rotation)_

*y*

*x*

*z*

## Navier-Stokes Equations    

- The Navier-Stokes equation govern fluid dynamics.  They consist of the
- statements of conservation of mass, momentum, and internal energy:
- The total internal energy is defined as
- The gravitational acceleration is defined by

## Navier-Stokes Equations (continued)   

- The total internal stress tensor is
- and includes the isotropic pressure component plus the viscous shear stress
- tensor. The **I** is the identity matrix. Assuming a Newtonian fluid in local
- thermodynamic equilibrium, the constitutive equation is
- This expression incorporates Stoke’s hypothesis that the bulk coefficient of
- viscosity is negligible for gases, and so, the second coefficient of viscosity is
- directly related to the coefficient of viscosity .

## Navier-Stokes Equations (continued)    

- The non-convective term of the flux can be separated into inviscid and
- viscous components in the form
- The heat flux vector can be expressed as
- and includes components of heat transfer due to molecular dissipation,
- diffusion of species, and radiation.

## Turbulence    

- When fluid dynamics exhibits instabilities, it is known as *turbulence*.  Most
- aerospace flows of interest involve the transition of flow from a laminar
- condition to turbulent.  Turbulence is defined as,
*“...**an irregular condition of flow in which the various quantities show a *
*random variation with time and space coordinates, so that statistically *
*distinct average values can be discerned” (Hinze, 1975).*
- Turbulence can be visualized as eddies (local swirling motion):
- Size of eddy is turbulent length scale, but smallest still larger than molecular length scale.
- Length scales vary considerably.
- Eddies overlap in space with large eddies carrying small eddies.
- Cascading process transfers the turbulent kinetic energy from large eddies to smaller eddies where energy is dissipated as heat energy through molecular viscosity.
- Eddies convect with flow, and so, turbulence is not local – depend on history of the eddy.

## Reynolds-Averaging    

- The fact that “*statistically distinct average values can be discerned*” allows us
- to apply a time-average over a time interval considered large with respect to
- the time scale of turbulence, but yet, small compared to the time scale of the
- flow.  We use Reynolds-averaging, which  replaces time-varying quantities
- with such relations as
- With similar expressions for (v,w,h,T).  This introduces Reynolds Stress
- into the time-averaged momentum and energy equations and the Reynolds
- Heat-Flux
- into the time-averaged energy equations.
*Task of turbulence modeling is to model these terms.*

## Simplifications of the RANS Equations    

- Simplification of the full Reynolds-Averaged Navier-Stokes (RANS) equations
- can be employed to allow less computational effort when certain flow physics
- is not to be simulated:
- _Thin-Layer Navier-Stokes Equations_
- In the case of thin boundary layers along a solid surface, one can neglect viscous terms
- for coordinate directions along the surface, which are generally small.
- _Parabolized Navier-Stokes (PNS) Equations_
- In addition to the thin-layer assumption, if the unsteady term is removed from the
- equation and flow is supersonic in the streamwise direction, then the equations become
- parabolic in the cross-stream coordinates and a space-marching method can be used,
- which reduces computational effort significantly.
- _Euler (Inviscid) Equations_
- If all viscous and heat conduction terms are removed from the RANS equations, the
- equations become hyperbolic in time.  This assumes viscosity effects are very small
- (high Reynolds numbers).

## Boundary and Initial Conditions    

- The Boundary Conditions for the Navier-Stokes equations will be
- discussed a little later.
- The type of Initial Conditions for the Navier-Stokes equations
- depend on the solution method.  Time-marching or iterative
- methods require an initial flow field solution throughout the
- domain.  Space-marching methods require the flow field solution
- at the starting marching surface.

## Turbulent Modeling    

- Turbulence Modeling is the process of closing the Navier-Stokes equations by
- providing required turbulence information.  Turbulence Modeling has a few
- fundamental classifications:
- _Models that use the Boussinesq Approximation_.  These are the *eddy-viscosity models*, which will be the focus of this presentation.
- _Models that solve directly for the Reynolds Stresses_.  These become complicated fast by introducing further terms requiring modeling.
- _Models not based on time-averaging_.  These are the Large-Eddy Simulation (LES) and Direct Numerical Simulation (DNS) methods.

## Boussinesq Approximation    

- The Boussinesq Approximation essentially assumes the exchange of turbulent energy in the cascading process of eddies is analogous of that of molecular viscosity.  Thus, the approximation is
- which is the same form as the laminar viscous tensor.  This allows us to write an
- effective viscosity as
- Similarly the Reynolds heat-flux vector is approximated by applying the
- Reynolds analogy between momentum and heat transfer
-            			   with			and
*Thus, the objective of the turbulence model is to compute ****T**.*

## Turbulence Equations    

- There exist various turbulence models, which may be classified as
- Algebraic Models
- Cebeci-Smith
- Baldwin-Lomax
- PDT
- One-Equation Models (1 pde)
- Baldwin-Barth
- Spalart-Allmaras (S-A)
- Two-Equation Models (2 pdes)
- SST
- Chien k-epsilon
- Models mostly assume fully turbulent flow rather than accurately model transition from laminar to turbulence flow.
- One- and two-equation models attempt to model the time history of turbulence.
- Models integrate through boundary layers to the wall, but S-A and SST models allow use of a wall function.

## Algebraic Models    

- Algebraic Models consist of algebraic relations to define the local eddy
- viscosity.  The models are based on Prandtl’s Mixing Length Model that was
- developed through an analogy with the molecular transport of momentum.
- The Cebeci-Smith Model is a two-layer model for wall-bounded flows that
- computes the turbulent eddy viscosity based on the distance from a wall using
- Prandtl’s model and empirical turbulence correlations.
- The Baldwin-Lomax Model improves on the correlations of the Cebeci-Smith
- model and does not require evaluation of the boundary layer thickness.  It is
- the most popular algebraic model.
- The PDT Model improves on the Baldwin-Lomax model for shear layers.
- Algebraic models work well for attached boundary layers under mild pressure
- gradients, but are not very useful when the boundary layer separates.

## One-Equation Models    

- One-Equation Models consist of a single partial differential equation (pde) that
- attempts to capture some of the history of the eddy viscosity.  The Baldwin-
- Barth and Spalart-Allmaras turbulence models are two popular models;
- however, the Baldwin-Barth model has some problems and its use is not
- recommended.
- The Spalart-Allmaras Model can be used when the boundary layer separates
- and has been shown to be a good, general-purpose model (at least robust to be
- used for a variety of applications).

## Two-Equation Models    

- Two-Equation Models consist of two partial differential equations (pde) that
- attempts to capture some of the history of the eddy viscosity.  The Menter SST
- and Chien k- turbulence models are two popular models.
- As the number of equations increases, the computational effort increases and
- one has to balance improvements in modeling with the capture of important
- turbulence information.

## Chemistry Equations    

- The chemistry equations govern multi-species diffusion and chemical reactions
- for a finite-rate fluids model.  The equations consist of species continuity
- equations:
- Where **j**i is the mass flux of species *i* defined by Fick’s first law of diffusion
- The rate of formation of species *i* is

## Chemistry Equations    

- The net rate of formation of species *i* is
- which is not zero for non-equilibrium flows.  For the set of chemical reactions
- specified for the fluid, the net rate of formation can be determined from
- where  denotes the product.  Again, note that

## Magneto-Fluid Dynamic Equations    

- The Magneto-Fluid Dynamics Equations allow the solution of
- the magnetic *B* and electric *E* fields.

## Normalization of the Equations    

- The equations can be normalized (non-dimensionalized) so that non-dimensional
- parameters (Reynolds and Prandtl number) are explicit in the equations and
- values are approximately unity.   Usual reference quantities are
-                                                        and
- with velocities non-dimensionalized by
- To allow       when      requires
- The non-dimensionalized equation of state becomes
- Where at reference (freestream) conditions
- This results in the relations

## Decoding the Conserved Variables    

- The solutions of the equations yields
-                                                                                                 for (*i *= 1, *ns-1*)
- Those variables are then “decoded” to determine other flow properties:
- A system of equations are iterated  on temperature *T* to determine *p*, *h*, and *T*
*	**	**	**	**	**	**	**(repeat)*

## Task From Here    

- We have defined the fluid relations and the governing
- conservation equations.  The task from here is to discretize
- the equations to allow a computational solution and apply
- numerical methods to obtain that solution.
- We start with looking at a finite-volume formulation of
- these equations.

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