1.3 Physics and Model Equations
The physics believed to govern all fluid flow on the length scale of interest in this project is classical Newtonian mechanics. That is, all length scales are assumed to be large enough to be able to consider the fluid as a continuum. This is in contrast to length scales where the individual molecules making up the fluid must be considered. A difference from classical mechanics is that for fluids one usually considers a velocity-field formulation instead of considering individual particles as is the usual approach in mechanics. This leads to a new form for the particle temporal derivative of a quantity, Q, for the fluid given (in 2D) as,
DQ Dt = ∂Q
∂t +u∂Q
∂x +v∂Q
∂y = ∂Q
∂t + (u· ∇)Q. (1.1) A full description and derivation of the difference between the particle and field approach may be found in [11, section 1-3].
1.3.1 General Navier-Stokes Equation
By applying Newtons second law of motion, PiFi =ma=mDuDt, to a "box of fluid" at an arbitrary position in an inertial frame of reference and dividing by the volume of the box one obtains a general form of the Navier-Stokes (NS) equations (1.2), which govern the motion of the fluid [11, section 2-4]:
ρ ∂u
∂t +u· ∇u
=−∇p+∇ ·τ +f. (1.2)
Here, u is the flow velocity, ρ the fluid density, p the pressure, τ a stress tensor and f represents external body forces acting on the fluid.
Each term in the NS-equation may be interpreted physically as follows.
The term: ∂u∂t. This term quite simply describes the change in the velocity of the fluid caused directly by a change in time. This term is clearly non-zero for all instationary problems and zero for stationary problems.
The term: u· ∇u. This term is called the convective acceleration term or the non-linear advection term. This term represents the fact that the fluid may be accelerated due to its position in the domain independently of time. That is, even for stationary problems (∂u∂t = 0) a fluid may change velocity throughout the domain. As an example of this one may consider a steady flow of incompressible liquid from one reservoir to another through a diverging channel. See figure 1.3 for an illustration of the channel.
u u u u
Figure 1.3: Illustration of a diverging open ended channel in which a steady flow of fluid will be fastest in the narrow end and slowest in the wide end.
This phenomenon is relatively easily understood if one considers mass conservation and continuity. Only the fluid which has passed through the narrow part of the channel is available to fill the channel down stream where the channel is wider. As the fluid must remain continuous the only choice available is to slow down as it moves down the channel.
The terms: −∇p+∇ ·τ. Both terms represent effects of stresses in the fluid. ∇pis the gradient of the pressure and stems from the normal stresses in the fluid. ∇ ·τ stems from the anisotropic part of the stresses and describes viscous forces. This can be understood as the friction forces between layers of fluid pulling on each other as they pass one another.
In order to simplify the NS-equations for the solution of engineering problems a series of assumptions can be made for τ but going into depth with this is outside the scope of this project. A simple assumption is that the fluid is incompressible which reducesτ as described in the following.
Further explanations of the terms and assumptions which may be made can be found in [11, chapter 2].
The term: f. This term represents all exterior forcing on the fluid. Normally, this always includes gravity and may also include forcings on the system under consideration from e.g.
electrical fields.
1.3.2 Incompressible Navier-Stokes Equation
For the problems of interest in this project the fluid under consideration is assumed in-compressible, temperature independent and it is assumed that the viscosity is constant throughout the fluid. These assumptions simplify the NS-equation to the form presented in equation 1.3,
ρ ∂u
∂t +u· ∇u
=−∇p+µ∇2u+f, (1.3)
It is easily seen that the change from the general NS-equation is the reduction of ∇ ·τ to µ∇2u which is denoted the viscous diffusion term. This term may be interpreted as modelling the diffusion of momentum which occurs for an incompressible viscous fluid.
1.3 Physics and Model Equations 1 THE PROBLEM AND UNDERLYING PHYSICS
The form of the NS-equation presented in (1.3) may be used directly as a component in solving an incompressible fluid problem. It is however beneficial to non-dimensionalize the NS-equation as this will reduce the parameters in the equation to a single parameter denoted the Reynolds number.
The first step is to introduce a mean flow velocityU and characteristic lengthL. From these a set of dimensionless functions and operators are then introduced as,
• Non-dimensional Velocity: uˇ = Uu.
• Non-dimensional Pressure: pˇ= ρUp2.
• Non-dimensional Body Force: ˇf= ρUL2f.
• New Time Derivative: ∂∂ˇt = UL∂t∂.
• New Spatial Derivative: ∇ˇ =L∇.
By multiplying (1.3) which is an equation in terms of body forces, with the quantity ρUL2, which has the unit ofhmN3i, one obtains the dimensionless equation,
L
By substituting in the dimensionless functions and operators one obtains,
∂
∂ˇtuˇ+ ˇu· ∇ˇu=−∇ˇˇp+ µ ρU L
∇ˇ2uˇ+ ˇf. (1.6)
Introducing the Reynolds number as, Re= ρU Lµ and dropping the check marks to simplify the notation one arrives at the final non-dimensional NS-equation,
∂
∂tu+u· ∇u=−∇p+ 1
Re∇2u+f. (1.7)
Note thatL=Dfor the model problems considered here.
1.3.3 Mass Conservation
Another requirement for a fluid problem is that of mass conservation. This principle simply states that mass cannot appear out of nothing and likewise can not disappear into nothing.
Thus, we must require,
m=ρV = constant, (1.8)
where m is the mass and V is the volume. For an incompressible fluid any "box of fluid"
must have a constant density in space and time. From this consideration and by taking the particle derivative of equation 1.8 one gets the condition,
Dm Dt = Dρ
DtV +ρDV
Dt = 0⇒ ∇ ·u= 0. (1.9) See [11, section 2-3] for a full derivation of this equation, i.e. for the missing steps hidden in the⇒.
(1.9) is easily non-dimensionalised by introducing,
• Non-dimensional Velocity: uˇ = Uu.
• New Spatial Derivative: ∇ˇ =L∇.
Multiplying (1.9) by LU and rewriting to obtain,
L∇ · u
U = ˇ∇ ·uˇ = 0. (1.10)
Dropping the check mark we get the non-dimensional equivalent of (1.9), which happens to have the exact same form.
1.3.4 Model Equations
We have now considered all the components needed to formulate a model for the problem.
The two main model equations are thus the non-dimensional NS- and continuity-equations:
∂
∂tu+u· ∇u=−∇p+ 1
Re∇2u+f, (1.11)
∇ ·u= 0. (1.12)
Additionally, we need to formulate boundary and initial conditions for the problem. Con-sidering the problem geometry described in section 1.2 a set of boundary conditions may be formulated.
Cylinder Boundary Condition: The boundary constituted by the cylinder surface uses the usual FD no-slip condition. For the stationary cylinder this may be formulated as:
u|(x,y)∈CS= (0,0), (1.13)
whereCS is the set of points which lie on the surface of the cylinder.
1.3 Physics and Model Equations 1 THE PROBLEM AND UNDERLYING PHYSICS
Wall Boundary: The boundary constituted by the moving wall also uses the usual FD no-slip condition, This may be formulated as:
u|(x,y)∈WS= (u∞,0), (1.14)
whereWS is the set of points which lie on the wall.
Far Field Boundaries: As the method of choice for solving the problem is simulation, and as it is impossible to simulate an infinite domain, one needs to approximate the infinite half plane above the wall by a finite box. The concerns in the reader’s mind for which this approximation gives rise will be addressed in section 3. The "box" approximation gives rise to two different far field BC’s. The first is an inflow condition given by:
u|(x,y)∈IF= (u∞,0), ∇p|(x,y)∈IF= 0. (1.15) where IF is the set of points which lie on the inflow boundaries. These BCs represent a completely undisturbed flow as one would expect far away from the cylinder5.
The second is an outflow condition given by;
∇u|(x,y)∈OF·n= (0,0), p|(x,y)∈OF=p∞= 0. (1.16) whereOF is the set of points which lie on the outflow boundary,nis the outward pointing normal vector and p∞ is a reference pressure which is set equal to zero. This BC simulates that the fluid simply flows freely out of the domain at the outflow boundary.
An illustration of the problem domain with the different BC’s highlighted is provided in figure 1.4.
Cylinder Outflow Wall Inflow
Figure 1.4: Illustration of the location of the different boundary conditions for the problem. (CS): Cylinder surface (full line), (WS): Wall surface (densely dotted),(IF): Inflow (densely dashed),(OF): Outflow (loosely dashed).
5Due to numerical performance the pressure condition stated here is not used in the solver from the Nek-tar++framework. Instead a higher order pressure condition derived to provide better numerical accuracy is used. Details of this condition and its derivation may be found in [18] and/or [16, section 8.3]
Initial Condition: The initial condition is chosen to be that of a free flow without the cylinder, i.e.
u|t=0 = (u∞,0), p|t=0=p∞= 0. (1.17) Since the investigation is not concerned with the initial transient solution, any reasonable choice of velocity and pressure fields as initial condition could be used as long as it is chosen to be compatible with the numerical method used.