This chapter consists of the objectives and the methodology of the thesis work. The reader is then introduced to the main content of the following chapters.
Chapter 2, discusses the theoretical background of basic equations describing fluid mo-tion. It explains how CFD formulates those equations. By using those equations, the Navier-Stokes equations are derived. In the middle part of this chapter discusses the principles of the CFD. The last part describes the definition of turbulence and its rich history.
Chapter 3 gives the brief details of turbulence models. The classification of the models are discussed based on filtering and time averaging. Different turbulence models such as the large eddy simulation (LES) and the Reynolds Averaged Navier-Stokes (RANS) models are explained with the suitability of each model in the applications of the flow past a circular cylinder and in a staggered tube bundle. The end part of this chapter contains the information about law of the wall.
Chapter 4 presents work done on the flow past a circular cylinder using RANS and LES methods at two different Reynolds numbers. The chapter focuses mainly on the verification and validation of RANS models and LES on the flow past a circular cylinder.
Pressure distribution, as well as the comparison of the Strouhal number and the drag coefficient of the flow from the prediction of RANS models and LES are compared to
experimental data. Grid and time-step size effect cases are discussed at the end also.
Chapter 5 contains work done on the flow past in a staggered tube bundle. The be-ginning part of this chapter displays the information about the staggered tube bundle.
The chapter focuses on the applicability and suitability of RANS models with present application and compared with experimental results. At the end, results obtained from CFD simulations for grid and time-step size influences are studied.
Chapter 6 draws conclusions of the whole thesis. This focuses on the objectives of the work and how they are achieved throughout the thesis.
2 Theoretical background
This chapter is a review of general theory of the governing equations for fluid flow. The governing equations of fluid flow are called the Navier-Stokes equations. In this section, concisely we will discuss the principles of the CFD with its components. Moreover, fundamental description of turbulence and its history will demonstrate.
2.1 Governing equations of fluid flow
In mid 18th century, the French engineer Claude Navier and the Irish mathematician George Stokes derived the well-known equations of fluid motion, known as the Navier-Stokes equations. These equations have been derived based on the fundamental govern-ing equations of fluid dynamics, called the continuity, the momentum and the energy equations, which represent the conservation laws of physics [15].
2.1.1 Continuity equation
Figure 1: (a) Fluid element for conservation laws (b) mass flows in and out of fluid element
In Figure 1a, the six faces are labeled N, S, E, W, T and B, which stands for North, South, East, West, Top and Bottom. The centre of the element is located at position (x, y, z). The derivation of the mass conservation equation is to write down a mass balance for the element:
Rate of increase of mass in fluid element = Net rate of flow of mass into fluid element [1]
The rate of increase of mass in the fluid element is
∂(ρδxδyδz)
∂t = ∂ρ
∂tδxδyδz (1)
Next we need to account flow rate across a face of the element, which is given by the product of density, area and the velocity component normal to the face. From Figure 1b it can be seen that the net rate of flow of mass into the element across its boundaries is given by
The rate of increase of mass inside the element from equation (1) is now equated to the net rate of flow of into the element across its faces from equation (2). All terms of the resulting mass balance are arranged on the left hand side of the equals sign and the expression is divided by the element volumeδxδyδz [2]. This yields
∂ρ
Equation (4) is the unsteady, three-dimensional mass conservation or continuity equation at a point in a compressible fluid [1].
For an incompressible fluid the densityρ is constant and equation (4) becomes
divu= 0 (5)
2.1.2 Momentum equation
Newton’s second law states that [1]:
Rate of increase of momentum of fluid particle = Sum of forces on fluid particle
The rates of increase of x-, y- and z-momentum per unit volume of a fluid particle are
Now, the state of stress of a fluid element is defined in terms of the pressure and the nine viscous stress components which is shown in Figure 2a.
Figure 2: (a) Stress components on three faces of fluid element (b) stress components in the x-direction
The pressure, a normal stress, is denoted by p. Viscous stresses are denoted by τ. The suffix notationτij is applied to indicate the direction of the viscous stresses. The sufficesi andj inτij indicate that the stress component acts in thej- direction on a surface normal to the i-direction. First we consider the x-components of the forces due to pressurep and stress componentsτxx,τyx andτzxwhich is shown in Figure 2b. Forces aligned with the direction of a co-ordinate axis get a positive sign and those in the opposite direction a negative sign. The net force in the x-direction is the sum of the force components acting in that direction on the fluid element.
On the pair of faces (E, W) we have The net force in the x-direction on the pair of faces (N, S) is
− And the net force in thex-direction on faces T and B is given by
− The total force per unit volume on the fluid due to these surface stresses is equal to the sum of equations (7), (8) and (9) divided by the volumeδxδyδz.
∂(−p+τxx)
∂x +∂τyx
∂y +∂τzx
∂z (10)
In addition, the body forces are not consider in the above explanation. In further detail their overall effect can be included by defining a source SM x of x-momentum per unit volume per unit time. Thex-component of the momentum equation is found by setting the rate of change ofx-momentum of the fluid particle in equation(6) equal to the total force in the x-direction on the element due to surface stresses in equation(10) plus the rate of increase ofx-momentum due to sources
ρDu
Dt = ∂(−p+τxx)
∂x +∂τyx
∂y + ∂τzx
∂z +SM x (11)
Similarly we can verify they-component of the momentum equation is given by
ρDv
Dt = ∂τxy
∂x +∂(−p+τyy)
∂y +∂τzy
∂z +SM y (12)
And thez-component of the momentum equation by
ρDw
Dt = ∂τxz
∂x +∂τyz
∂y +∂(−p+τzz)
∂z +SM z (13)
The source terms SM x, SM y and SM z in above equations include contributions due to body forces only.
2.1.3 Navier-Stokes equations for a Newtonian fluid
The most useful forms of the conservation equation for fluid flows are obtained by in-troducing a suitable model for the viscous stresses τij. In many fluid flows the viscous
stresses can be expressed as functions of the local deformation rate or strain rate [1]. In three dimensional flows, the local rate of deformation is composed of the linear defor-mation rate and the volumetric defordefor-mation rate. The rate of linear defordefor-mation of a fluid element has nine components in three dimensions, six of which are independent in isotropic fluid. they are denoted by the symbolsij.
In a Newtonian fluid the viscous stresses are proportional to the rates of deformation [2].
The three dimensional form of Newton’s law of viscosity for compressible flows involves two constants of proportionality: 1) dynamic viscosity, µ, to relate stresses to linear deformation, and 2) viscosity, λ, to relate stresses to the volumetric deformation. The viscous stress components, of which six are independent, are
τxx= 2µ∂u The second viscosity λ is not known much because of its effect is small in practice.
Substitution of the above shear stresses equation(14) into equations(11), (12) and (13) yields the so-called Navier-Stokes equations.
Now, rearrange the viscous stress terms as follows:
the viscous stresses in they- andz-component equations can be rearrange in a similar manner. And defining a new source terms by
SN =SM + [sM] (18)
Finally the Navier-Stokes equations can be written in the most useful form is ρDu Here the source termsSN x,SN y and SN z in above equations include contributions due to body forces. By solving these equations, the pressure and velocity of the fluid can be predicted throughout the flow.