2. FLUIDFLOW EQUATIONS SPRING 2019


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1 2. FLUIDFLOW EQUATIONS SPRING Introduction 2.2 Conservative differential equations 2.3 Nonconservative differential equations 2.4 Nondimensionalisation Summary Examples 2.1 Introduction Fluid dynamics is governed by conservation equations for: mass; momentum; energy; (for a nonhomogenous fluid) other constituents. Equations for these can be expressed mathematically as: integral (controlvolume) equations; differential equations. This course focuses on the controlvolume approach (the basis of the finitevolume method) because it relates naturally to physical quantities, is intrinsically conservative and is easier to apply in modern, unstructuredmesh CFD with complex geometries. However, the equivalent differential equations are easier to write down, manipulate and, in a few cases, solve analytically. Although there are many different physical quantities, most satisfy a single generic equation: the scalartransport or advectiondiffusion equation. V (1) The finitevolume method is a natural discretisation of this. CFD 2 1 David Apsley
2 2.2 Conservative Differential Equations Mass Conservation (Continuity) Mass is neither created nor destroyed, so: rate of change of mass in cell = net inward mass flux With the more conventional flux direction (positive outward): rate of change of mass in cell + net outward mass flux = 0 (2) For a cell volume V and a typical face area A: mass of fluid in the cell: mass flux through one face: V A u u n t y w z s b n e A conservative differential equation for mass conservation can be derived by considering the small Cartesian control volume shown left x If density and velocity are averages over cell volume or cell face, respectively: Writing and A w = A e = ΔyΔz etc: Dividing by the volume, xyz: i.e. Taking the limit as Δx, Δy, Δz 0: (3) CFD 2 2 David Apsley
3 This analysis is analogous to the finitevolume procedure, but there the control volume does not shrink to a point (finitevolume, not infinitesimalvolume) and cells can be any shape. (*** Advanced / Optional ***) For an arbitrary volume V with closed surface V: (4) For a fixed control volume, take d/dt under the integral sign and apply the divergence theorem to turn the surface integral into a volume integral: Since V is arbitrary, the integrand must be identically zero. Hence, (5) Incompressible Flow For incompressible flow, volume as well as mass is conserved, so that: Substituting for face areas, dividing by volume and proceeding to the limit as above produces (6) This is usually taken as the continuity equation in incompressible flow. Note that it is irrelevant whether the flow is timedependent or not. CFD 2 3 David Apsley
4 2.2.2 Momentum Newton s Second Law: rate of change of momentum = force rate of change of momentum in cell + net outward momentum flux = force (7) For a cell volume V and a typical face area A: momentum of fluid in the cell = mass u momentum flux through a face = V A u u n Momentum and force are vectors, giving (in principle) 3 equations. Fluid Forces There are two main types: surface forces (proportional to area; act on controlvolume faces) body forces (proportional to volume) (i) Surface forces are usually expressed in terms of stress: or The main surface forces are: pressure p: acts normal to a surface; viscous stresses τ: frictional forces arising from relative motion. y For a simple shear flow there is only one nonzero stress component: but, in general, τ ij is a symmetric tensor with a more complex expression for its components. In incompressible flow 1, 22 U 12 or, for a general component, y x There is a slightly extended expression in compressible flow; see the recommended textbooks. CFD 2 4 David Apsley
5 (ii) Body forces are often expressed as forces per unit volume, or force densities. The main body forces are: gravity: the force per unit volume is z g (For constantdensity fluids, pressure and weight can be combined as a piezometric pressure p* = p + ρgz; gravity then no longer appears explicitly in the flow equations.) centrifugal and Coriolis forces (apparent forces in a rotating reference frame): axis centrifugal force: R 2 R r Coriolis force: u In inertial frame In rotating frame Because the centrifugal force can be written as the gradient of some quantity in this case it can also be absorbed into a modified pressure and removed from the momentum equation; see the Examples. t Differential Equation For Momentum Consider a fixed Cartesian control volume with sides Δx, Δy, Δz. Follow the same process as for mass conservation. y w z s b n e For the xcomponent of momentum: x Substituting cell dimensions: Dividing by volume ΔxΔyΔz (and changing the order of p e and p w ): CFD 2 5 David Apsley
6 In the limit as Δx, Δy, Δz 0: (8) Notes. (1) The viscous term is given without proof (but see the notes below). 2 is the Laplacian operator. (2) The pressure force per unit volume in any direction is minus the pressure gradient in that direction. (3) The y and zmomentum equations can be obtained by inspection / patternmatching. (*** Advanced / Optional ***) With surface forces determined by stress tensor σ ij and body forces determined by force density f i, the controlvolume equation for the i component of momentum may be written (9) For fixed V, take d/dt inside integrals and apply the divergence theorem to surface integrals: As V is arbitrary, the integrand vanishes identically. Hence, for arbitrary forces: (10) The stress tensor has pressure and viscous parts: (11) (12) For a Newtonian fluid, the viscous stress tensor (including compressible part) is given by If the fluid is incompressible and viscosity is uniform then the viscous term simplifies to give CFD 2 6 David Apsley
7 2.2.3 General Scalar A similar equation may be derived for any physical quantity that is advected and diffused in a fluid flow. Examples include salt, sediment and chemical pollutants. For each such quantity an equation is solved for the concentration (amount per unit mass of fluid). Diffusion causes net transport from regions of high concentration to regions of low concentration. For many scalars this rate of transport is proportional to area and concentration gradient and may be quantified by Fick s diffusion law: This is often referred to as gradient diffusion. An example is heat conduction. For an arbitrary control volume: amount in cell: ρv (mass concentration) advective flux: (mass flux concentration) V A u u n diffusive flux: ( diffusivity gradient area) source: S = sv (source density volume) Balancing the rate of change, the net flux through the boundary and rate of production yields the general scalartransport (or advectiondiffusion) equation: (13) (Conservative) differential equation: (14) (*** Advanced / Optional ***) The integral equation may be expressed more mathematically as: For a fixed control volume, taking the time derivative under the integral sign and using the divergence theorem gives a corresponding conservative differential equation: (15) (16) CFD 2 7 David Apsley
8 2.2.4 Momentum Components as Transported Scalars In the momentum equation, if the viscous force looks like a diffusive flux. For example, for the xcomponent: is transferred to the LHS it Compare this with the generic scalartransport equation: Each component of momentum satisfies its own scalartransport equation, with concentration, velocity component (u, v or w) diffusivity, Γ viscosity μ source, S other forces Consequently, only one generic scalartransport equation need be considered. In Section 5 we shall see, however, that the momentum components differ from passive scalars (those not affecting the flow), because: equations are nonlinear (mass flux involves the velocity component being solved for); equations are coupled (mass flux involves the other velocity components as well); the velocity field must also be massconsistent NonGradient Diffusion The analysis above assumes that all nonadvective flux is simple gradient diffusion: Actually, the real situation is more complex. For example, in the umomentum equation the full expression for the 1component of viscous stress through the 2face is The u/ y part is gradient diffusion of u, but the v/ x term is not. In general, nonadvective fluxes that can t be represented by gradient diffusion are discretised conservatively (i.e. worked out for cell faces, not particular cells), then transferred to the RHS as a source term: CFD 2 8 David Apsley
9 2.2.6 Moving Control Volumes Controlvolume equations are also applicable to moving control volumes, provided the normal velocity component in the mass flux is that relative to the mesh; i.e. The finitevolume method can thus be used for calculating flows with moving boundaries NonConservative Differential Equations Conservative differential equations are socalled because they can be integrated directly to give an equivalent integral form involving the net change in a flux, with the flux leaving one cell equal to that entering an adjacent cell. To do so, all terms involving derivatives of dependent variables must have differential operators on the outside. In one dimension: (*** Advanced / Optional ***) The threedimensional version uses partial derivatives and the divergence theorem to change the differentials to surface flux integrals. As an example of how the same equation can appear in conservative and nonconservative forms, consider a simple 1d example: (conservative form can be integrated directly) (nonconservative form, obtained by applying the chain rule) Material Derivatives The time rate of change of some property in a fluid element moving with the flow is called the material (or substantive) derivative. It is denoted by D/Dt and defined below. Every field variable is a function of both time and position; i.e. As one follows a path through space, changes with time because: it changes with time t at each point; and it changes with position (x, y, z) as it moves with the flow. (x(t), y(t), z(t)) 2 See, for example: Apsley, D.D. and Hu, W., 2003, CFD Simulation of two and threedimensional freesurface flow, International Journal for Numerical Methods in fluids, 42, CFD 2 9 David Apsley
10 Thus, the total time derivative following an arbitrary path (x(t), y(t), z(t)) is The material derivative is the time derivative along the particular path following the flow (dx/dt = u, etc.): or (17) In particular, the material derivative of velocity (Du/Dt) is the acceleration (Hydraulics 1). For general scalar, the sum of timedependent and advection terms (total rate of change) is (by the product rule) (18) Using the material derivative, a scalartransport equation can thus be written in a much more compact, but nonconservative, form. In particular, the momentum equation becomes (19) This form is much simpler to write and is used both for convenience and to derive theoretical results in special cases (see the Examples). However, in the finitevolume method it is the longer, conservative form which is actually discretised. CFD 2 10 David Apsley
11 (*** Advanced / Optional ***) The derivation of (18) above is greatly simplified by use of the summation convention: or vector derivatives: Alternatively, conservative differential equations may derived from fixed control volumes (Eulerian approach) and their nonconservative counterparts from control volumes moving with the flow (Lagrangian approach). CFD 2 11 David Apsley
12 2.4 NonDimensionalisation Although it is possible to work entirely in dimensional quantities, there are good theoretical reasons for working in nondimensional variables. These include the following. All dynamicallysimilar problems (same Re, Fr etc.) can be solved with a single computation. The number of relevant parameters (and hence the number of graphs needed to report results) is reduced. It indicates the relative size of different terms in the governing equations and, in particular, which might conveniently be neglected. Computational variables are of a similar order of magnitude (ideally, of order unity), yielding better numerical accuracy NonDimensionalising the Governing Equations For incompressible flow the governing equations are: continuity: (20) momentum: (21) (and similar in y, z directions) Adopting reference scales U 0, L 0 and ρ 0 for velocity, length and density, respectively, and derived scales L 0 /U 0 for time and for pressure, each fluid quantity can be written as a product of a dimensional scale and a nondimensional variable (indicated by a *): Substituting into mass and momentum equations (20) and (21) yields, after simplification: (22) where (23) From this, it is seen that the key dimensionless group is the Reynolds number Re. If Re is large then viscous forces would be expected to be negligible in much of the flow. Having derived the nondimensional equations it is usual to drop the asterisks and simply declare that you are working in nondimensional variables. CFD 2 12 David Apsley
13 Note. The objective is that nondimensional quantities (e.g. p*) should be order of magnitude unity, so the scale for a quantity should reflect its range of values, not necessarily its absolute value. In incompressible (but not compressible) flow it is differences in pressure that are important, not absolute values. Since flowinduced pressures are usually much smaller than the absolute pressure, one usually works with departures from a constant reference pressure p ref and use as a scale magnitude; Hence, we nondimensionalise as:. Similarly, in Section 3 when we look at small changes in density due to temperature or salinity that give rise to buoyancy forces we shall use an alternative nondimensionalisation: with Δρ the overall size of density variation and θ* typically varying between 0 and 1. (24) Common Dimensionless Groups If other types of fluid force are included then each introduces another nondimensional group. For example, gravitational forces lead to a Froude number (Fr) and Coriolis forces to a Rossby number (Ro). Some of the most important dimensionless groups are given below. U and L are representative velocity and length scales, respectively. Reynolds number (viscous flow; μ = dynamic viscosity) Froude number (openchannel flow; g = gravity) Mach number (compressible flow; c = speed of sound) Rossby number (rotating flows; Ω = angular velocity of frame) Weber number (interfacial flows; σ = surface tension) Note. For flows with buoyancy forces caused by a change in density, rather than openchannel flows, we sometimes use a densimetric Froude number instead; this is defined by Here, g is replaced in the usual formula for Froude number by reduced gravity g: see Section 3., sometimes called the CFD 2 13 David Apsley
14 Summary Fluid dynamics is governed by conservation equations for mass, momentum, energy (and, for a nonhomogeneous fluid, the amount of individual constituents). The governing equations can be written in equivalent integral (controlvolume) or differential forms. The finitevolume method is a direct discretisation of the controlvolume equations. Differential forms of the flow equations may be conservative (i.e. can be integrated directly to something of the form flux out flux in = source ) or nonconservative. For any conserved quantity and arbitrary control volume: rate of change + net outward flux = source There are really just two canonical equations to discretise and solve: mass conservation (continuity): scalartransport (or advectiondiffusion) equation: Each velocity component (u, v, w) satisfies its own scalartransport equation. However, these equations differ from those for a passive scalar because they are nonlinear, coupled through the advective fluxes and pressure forces, and the velocity field is also required to be massconsistent. Nondimensionalising the governing equations allows dynamicallysimilar flows (those with the same values of Reynolds number, etc.) to be solved with a single calculation, reduces the overall number of parameters, indicates whether certain terms in the governing equations are significant or negligible and ensures that the main computational variables are of similar magnitude. CFD 2 14 David Apsley
15 Examples Q1. In 2d flow, the continuity and xmomentum equations can be written in conservative form as respectively. (a) Show that these can be written in the equivalent nonconservative forms: where the material derivative is given (in 2 dimensions) by. (b) (c) (d) (e) (f) Define carefully what is meant by the statement that a flow is incompressible. To what does the continuity equation reduce in incompressible flow? Write down conservative forms of the 3d equations for mass and xmomentum. Write down the zmomentum equation, including the gravitational force. Show that, for constantdensity flows, pressure and gravity forces can be combined in the momentum equations via the piezometric pressure p + ρgz. In a rotating reference frame there are additional apparent forces (per unit volume): centrifugal force: Coriolis force: where is the angular velocity of the reference frame, u is the fluid velocity in that frame, r is the position vector (relative to a point on the axis of rotation) and R is its projection perpendicular to the axis of rotation. ( denotes vector product.) By writing the centrifugal force as the gradient of some quantity show that it can be subsumed into a modified pressure. Also, find the components of the Coriolis force if rotation is about the z axis. or axis R r 2 R (*** Advanced / Optional ***) (g) (h) Write the conservative mass and momentum equations in vector notation. Write the conservative mass and momentum equations in suffix notation using the summation convention. CFD 2 15 David Apsley
16 Q2. (Exact solutions of the NavierStokes equation) The xcomponent of the momentum equation is given by Using this equation derive the velocity profile in fullydeveloped, laminar flow for: (a) pressuredriven flow between stationary parallel planes ( Plane Poiseuille flow ); (b) constantpressure flow between stationary and moving planes ( Couette flow ). Assume flow in the x direction, with bounding planes y = 0 and y = h. The velocity is then (u(y),0,0). In part (a) both walls are stationary. In part (b) the upper wall slides parallel to the lower wall with velocity U w. y x u(y) h (c) (*** Advanced / Optional ***) In cylindrical polars (x,r,) the Laplacian 2 is more complicated. If axisymmetric, with fullydeveloped velocity, then Derive the velocity profile in a circular pipe with stationary wall at r = R ( Poiseuille flow ). Q3. (*** Advanced / Optional ***) By applying the divergence theorem, deduce the conservative and nonconservative differential equations corresponding to the general integral scalartransport equation Q4. In each of the following cases, state which of (i), (ii), (iii) is a valid dimensionless number. Carry out research to find the name and physical significance of these numbers. (L = length; U = velocity; z = height; p = pressure; ρ = density; μ = dynamic viscosity; ν = kinematic viscosity; g = gravitational acceleration; Ω = angular velocity). (a) (i) (ii) (iii) (b) (i) (ii) (iii) (c) (i) (ii) (iii) (d) (i) (ii) (iii) CFD 2 16 David Apsley
17 Q5. (Exam 2008; part (h) depends on later sections of this course) The momentum equation for a viscous fluid in a rotating reference frame is (*) where ρ is density, u = (u,v,w) is velocity, p is pressure, μ is dynamic viscosity and angularvelocity vector of the reference frame. The symbol denotes a vector product. is the (a) If write down the x and y components of the Coriolis force ( ). (b) Hence write down the x and ycomponents of equation (*). (c) (d) (e) Show how Equation (*) can be written in nondimensional form in terms of a Reynolds number Re and Rossby number Ro (both of which should be defined). Define the terms conservative and nonconservative when applied to the differential equations describing fluid flow. Define (mathematically) the material derivative operator D/Dt. Then, noting that the continuity equation can be written show that the xmomentum equation can be written in an equivalent conservative form. (f) (g) (h) If the xmomentum equation were to be regarded as a special case of the general scalartransport (or advectiondiffusion) equation, identify the quantities representing: (i) concentration; (ii) diffusivity; (iii) source. Explain why the three equations for the components of momentum cannot be treated as independent scalar equations. Explain (briefly) how pressure is determined in a CFD simulation of: (i) highspeed, compressible gas flow; (ii) incompressible flow. CFD 2 17 David Apsley
18 Q6. (a) In a rotating reference frame (with angular velocity vector ) the nonviscous forces on a fluid are, per unit volume, where p is pressure, g = (0,0, g) is the gravity vector and R is the vector from the closest point on the axis of rotation to a point. Show that, in a constantdensity fluid, force densities (I), (II) and (III) can be combined in terms of a modified pressure. (b) Consider a closed cylindrical can of radius 5 cm and depth 15 cm. The can is completely filled with fluid of density 1100 kg m 3 and is rotating steadily about its axis (which is vertical) at 600 rpm. Where do the maximum and minimum pressures in the can occur, and what is the difference in pressure between them? Q7. (Exam 2011 part) The figure below depicts a 2d cell in a finitevolume CFD calculation. Vertices are given in the figure, and velocity in the adjacent table. At this instant ρ = 1.0 everywhere. (a) (b) Calculate the volume flux out of each face. (Assume unit depth.) Show that the flow is not incompressible and find the time derivative of density. y (1,2) w x (0,0) s n (4,0) e (6,2) Face Velocity (u,v) u v e 4 10 n 1 8 w 2 2 s 1 4 CFD 2 18 David Apsley
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