#### Introduction

ColdStream supports 3 types of cases:

- Custom designs
- Standard designs
- Simulations

The first two options are optimizations, where ColdStream will return an optimized design. The third option is to be used of you already have a design and you would like to know its performance.

This document describes the modelling approach used within ColdStream to perform CFD simulations. The equations used to model the fluid and solid regions physical behavior are explained. The different turbulence models, used to describe the turbulent behavior of fluids, available in ColdStream are also described.

#### Fluid region

__Navier-Stokes equations__

In coldStream, the physical behavior of fluids is modeled using the Navier-Stokes equations, conservation of mass and momentum, in combination with the energy equation. [1], [2]

For incompressible fluids, the continuity equation can be written as:
\[\frac{\partial\rho}{\partial t} + \nabla \cdot \rho \vec{v} = \rho \nabla \cdot \vec{v} = 0\]
Where \(\rho\) represents the density field, \(\vec{v}\) is the velocity field and \(t\) is the time.

The incompressible flow assumption holds well for all fluids at low Mach numbers (up to approximately 0.3). For higher Mach numbers, the compressibility effects cannot be ignored and should be taken into account. The Mach number for gas flows can be calculated using the following equation (liquids are always assumed to be incompressible):
\[M=\frac{v}{c}=\frac{v}{\sqrt{\gamma\space R\space T}}\]
Where \(v\) is the velocity magnitude, \(c\) represents the speed of sound, \(\gamma\) is the heat capacity ratio, \(R\) is the specific gas constant and \(T\) is the temperature.

The conservation of momentum can be written as:
\[\rho \frac{\partial \vec{v}}{\partial t} + \rho \vec{v} \cdot \nabla \vec{v} = -\nabla p + \nabla \cdot \mu \nabla\vec{v} + \vec{f}\]
Where \(\vec{v}\) is the velocity field, \(\rho\) is the density, \(\mu\) is the dynamic viscosity, \(p\) is the pressure and \(\vec{f}\) an optional body force (for example gravity).

The energy equation, as a function of enthalpy (\(h\)), can be written as: \[\rho \frac{\partial h}{\partial t} + \rho \vec{v} \cdot \nabla h - \rho \nabla\cdot \alpha \nabla h = Q\] Where \(\alpha\) is the thermal diffusivity and \(Q\) is an energy source (for example an internal heat source due to an exothermic reaction). The thermal diffusivity \(\alpha\) is defined as: \[\alpha = \frac{k}{\rho c_{p}}\] Where \(k\) represents the thermal conductivity and \(c_p\) the specific heat capacity. The following equation relates the ethalpy to the temperature for incompressible fluids: \(dh = c_p \space dT\). The energy equation can therefore be easily written in terms of the temperature: \[\rho c_p \frac{\partial T}{\partial t} + \rho c_p \vec{v} \cdot \nabla T - \nabla\cdot k \nabla T = Q\]

__Body forces__

For forced convection applications, \(\vec{f} = \vec{0}\). For natural convection applications, the body Boussinesq approximation is applied to model the buoyant behavior of the fluid. Therefore, the body force is taken as: \[\vec{f} = \rho (1 - \beta (T - T_{ref} ))\vec{g}\] Where \(\vec{g}\) is the gravitational acceleration, \(\beta\) the coeficient of expansion. Note that \(\beta\) is taken as a function of the reference temperature (\(T_{ref}\)) and therefore is constant during the simulation.

__Turbulence model__

Different modeling approaches exist to predict the effects of turbulence on a flowing fluid. An overview of the available turbulence models withing ColdStream and a flow chart to help you select the best model for your specific problem can be found here.

#### Solid region

For the solid region, only the energy equation needs be solved: \[\rho \frac{\partial h}{\partial t} - \rho \nabla\cdot \alpha \nabla h = Q\] Where \(\rho\) is the density of the solid material, \(h\) is the enthalpy, \(\alpha\) represents the thermal diffusivity and \(Q\) is a heat source (for example due to Joule heating). Similarly to the fluid equations, the solid energy equation can also be written in terms of the temperature (\(T\)): \[\rho\space c_p\space\frac{\partial T}{\partial t} - \nabla\cdot k \nabla T = Q\] Where \(c_p\) is the specific heat capacity and \(\kappa\) represents the thermal conductivity of the solid material. ColdStream allows to use either isotropic or anisotropic properties, depending on the material thermophysical properties.

#### Interface modeling

The heat exchange between different components, for example a solid and a fluid, depends only on the temperature difference between the respective cells adjacent to the wall. At this wall, interface equilibrium conditions are imposed, requiring that the heat flux over the interface is equal and from fluid side only dependent on the effective thermal conductivity, which accounts for the thermal conductivity of the fluid and the turbulent diffusivity: \[\kappa_{solid}\space\nabla T_{solid}=\kappa_{fluid}\space\nabla T_{fluid}\] Furthermore, the temperature on each side of the interface between a fluid and a solid should match: \[T_{solid} = T_{fluid}\] While computationally expensive, this approach provides a detailed insight into the physics of the problem, which in turn, will provide the optimizer with a more realistic performance of each design. This is fundamental in order to correctly explore the design space and eventually find a true optimal design.

#### References

- https://www.eng.auburn.edu/~tplacek/courses/fluidsreview-1.pdf
- https://www.grc.nasa.gov/www/k-12/airplane/nseqs.html