# Numerical Method

Licentiate Thesis / Nenad Glodic

6 NUMERICAL INVESTIGATIONS 6.1 Numerical Method

Numerical simulations are carried out employing a commercial CFD code (ANSYS CFX v11). The solver is using a full-scale time-marching 3D viscous model. Underlying equations, three dimensional Navier-Stokes equations in their conservation form, are being solved by using a Finite Volume method, where equations are integrated over the finite control volumes. Thereby, the solution domain is subdivided into a finite number of control volumes employing a suitable grid, which defines the control boundaries around a computational node in each control volume center. 6.1.1 Governing equations In fluid dynamics, the fluid flow is governed by the conservation laws for mass, momentum and energy. The basic conservation laws are formulated by using Leibniz-Reynolds transport theorem, which is an integral relation stating that the changes of some intensive property defined over a control volume must be equal to what is lost (or gained) through the boundaries of the volume plus what is created/consumed by sources and sinks inside the control volume. The threedimensional conservation equations (mass conservation, x-,y- and z-momentum equations and energy equation) for compressible fluid can be given in differential form (Versteeg & Malalasekera, 1995) as follows u 0 t Du p xx yx zx S Mx Dt x y z Dv xy p yy zy S My Dt x y z Dv xz yz p zz S Mz Dt x z z u xx u yx u zx y z x v v v DE xy yy zy u kT S E Dt y z x w xz w yz w zz x y z

Eq. 6-1 Eq. 6-2 Eq. 6-3 Eq. 6-4

Eq. 6-5

where the source terms S Mx , S My , S Mz include contributions due to body forces only. The specific energy of the fluid E in Eq.6-5 is defined as the sum of internal 1 (thermal) energy i and kinetic energy u 2 v 2 w 2 , while the effects of 2 gravitational potential energy changes are included as energy source term

Licentiate Thesis / Nenad Glodic

Page 49

(gravitational force regarded as a body force). It is common practice to extract the changes of the kinetic energy in order to obtain an equation for internal energy i . This is done by taking the part of energy attributable to the kinetic energy, which can be obtained by multiplying the momentum equations with corresponding velocity components and adding them together, and subtracting it from the above presented energy equation (Eq.6-5) xx Di p u Dt zy u u yx zx x y v w xz yz z x u xy z w zz y v v yy x y kT S i w z

Eq.6-6

Among the unknowns in the above presented conservation equations are four thermodynamic variables: , p, i and T . The relationship between these thermodynamic variables can be obtained through the assumption of thermodynamic equilibrium, where the state of substance in the equilibrium can be described by means of just two state variables. This yields, for a perfect gas, the well-known equations of state: p RT and i cv T

Eq. 6-7

In the flow of compressible fluids the equations of state provide the linkage between the energy equation on one hand and mass conservation and momentum equations on the other. The governing equations also contain the viscous stress components ij , and in a Newtonian fluid the viscous stresses are proportional to rates of deformation or strain rate. For compressible flow, Newton’s law of viscosity involves two constants of proportionality: dynamic viscosity , to relate stresses to deformation, and second viscosity , to relate stresses to volumetric deformation. The second viscosity is very difficult to determine and is often neglected. The shear stress components can...

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