Lecture 3 - Conservation Equations Applied Computational Fluid Dynamics Instructor: André Bakker http://www.bakker.org © André Bakker (2002-2006) 1 Governing equations • The governing equations include the following conservation laws of physics: – Conservation of mass. – Newton’s second law: the change of momentum equals the sum of forces on a fluid particle. – First law of thermodynamics (conservation of energy): rate of change of energy equals the sum of rate of heat addition to and work done on fluid particle. • The fluid is treated as a continuum. For length scales of, say, 1µm and larger, the molecular structure and motions may be ignored. 2 Lagrangian vs. Eulerian description A fluid flow field can be thought Another view of fluid motion is of as being comprised of a large the Eulerian description. In the number of finite sized fluid Eulerian description of fluid particles which have mass, motion, we consider how flow momentum, internal energy, and properties change at a fluid other properties. Mathematical element that is fixed in space laws can then be written for each and time (x,y,z,t), rather than fluid particle. This is the following individual fluid Lagrangian description of fluid particles. motion. Governing equations can be derived using each method and converted to the other form. 3 Fluid element and properties Fluid element for conservation laws • The behavior of the fluid is described in terms of macroscopic properties: – Velocity u. – Pressure p. δδδ (x,y,z) z – Density ρ. – Temperature T. δδδy – Energy E. δδδx z • Typically ignore (x,y,z,t) in the notation. y • Properties are averages of a sufficiently x large number of molecules. Faces are labeled • A fluid element can be thought of as the North, East, West, smallest volume for which the continuum South, Top and Bottom assumption is valid. Properties at faces are expressed as first two terms of a Taylor series expansion, ∂p1 ∂ p 1 e.g.forp:pp=−δ x and pp =+ δ x W∂x2 E ∂ x 2 4 Mass balance • Rate of increase of mass in fluid element equals the net rate of flow of mass into element. ∂ ∂ρ • Rate of increase is: (ρδ xδyδz) = δxδyδz ∂t ∂t • The inflows (positive) and outflows (negative) are shown here: ∂(ρw ) 1 ρw+ . δδδ zxy ∂z 2 ∂(ρv ) 1 ρv+ . δδδ yxz ∂y 2 ∂(ρu ) 1 ∂(ρu) 1 ρu+ . δδδ xyz ρu − . δx δyδz ∂x 2 ∂x 2 ∂(ρv) 1 ρv − . δy δxδz z ∂y 2 y ∂(ρw) 1 x ρw − . δz δxδy ∂z 2 5 Continuity equation • Summing all terms in the previous slide and dividing by the volume δxδyδz results in: ∂ρ ∂ ρ ∂ ρ ∂ ρ + ( u) + ( v) + ( w) = 0 ∂t ∂x ∂y ∂z • In vector notation: ∂ρ + div (ρ u) = 0 ∂t Change in density Net flow of mass across boundaries Convective term • For incompressible fluids ∂ρ /∂ t = 0, and the equation becomes: div u = 0. ∂ ∂ ∂ ∂u u + v + w = i = • Alternative ways to write this: ∂ ∂ ∂ 0 and ∂ 0 x y z xi 6 Different forms of the continuity equation Finite control volume Finite control volume fixed fixed in space mass moving with flow ∂ D ∫∫∫ ρ dV + ∫∫ ρ U ⋅dS = 0 ∫∫∫ ρ dV = 0 ∂t V S Dt V Integral form Integral form Conservati on form Non − conservati on form U Infinitesimally small Infinitesimally small fluid element of fixed element fixed in space mass (“fluid particle”) moving with the flow ∂ρ Dρ + ∇ ⋅ (ρU) = 0 + ρ∇ ⋅ U = 0 ∂t Dt Differenti al form Differenti al form Conservati on form Non − conservati on form 7 Rate of change for a fluid particle • Terminology: fluid element is a volume stationary in space, and a fluid particle is a volume of fluid moving with the flow. • A moving fluid particle experiences two rates of changes: – Change due to changes in the fluid as a function of time. – Change due to the fact that it moves to a different location in the fluid with different conditions. • The sum of these two rates of changes for a property per unit mass φ is called the total or substantive derivative Dφ /Dt : φ ∂φ ∂φ ∂φ ∂φ D = + dx + dy + dz Dt ∂t ∂x dt ∂y dt ∂z dt • With dx/dt=u , dy/dt=v , dz/dt=w , this results in: Dφ ∂φ = + u.grad φ Dt ∂t 8 Rate of change for a stationary fluid element • In most cases we are interested in the changes of a flow property for a fluid element, or fluid volume, that is stationary in space. • However, some equations are easier derived for fluid particles. For a moving fluid particle, the total derivative per unit volume of this property φ is given by: Dφ ∂φ (for moving fluid particle) ρ = ρ + u.grad φ (for given location in space) Dt ∂t • For a fluid element, for an arbitrary conserved property φ: ∂ρ ∂(ρφ ) + div (ρ u) = 0 + div (ρφ u) = 0 ∂t ∂t Continuity equation Arbitrary property 9 Fluid particle and fluid element • We can derive the relationship between the equations for a fluid particle (Lagrangian) and a fluid element (Eulerian) as follows: ∂(ρφ ) ∂φ ∂ρ Dφ + div (ρφ u) = ρ + u.grad φ + φ + div (ρu) = ρ ∂t ∂t ∂t Dt zero because of continuity ∂(ρφ ) Dφ + div (ρφ u) = ρ ∂t Dt Rate of increase of Net rate of flow of Rate of increase of = φφφ of fluid element φφφ out of fluid element φφφ for a fluid particle 10 To remember so far • We need to derive conservation equations that we can solve to calculate fluid velocities and other properties. • These equations can be derived either for a fluid particle that is moving with the flow (Lagrangian) or for a fluid element that is stationary in space (Eulerian). • For CFD purposes we need them in Eulerian form, but (according to the book) they are somewhat easier to derive in Lagrangian form. • Luckily, when we derive equations for a property φ in one form, we can convert them to the other form using the relationship shown on the bottom in the previous slide. 11 Relevant entries for Φ x-momentum u Du ∂(ρu) ρ + div (ρuu) Dt ∂t y-momentum v Dv ∂(ρv) ρ + div (ρvu) Dt ∂t z-momentum w Dw ∂(ρw) ρ + div (ρwu) Dt ∂t Energy E DE ∂(ρE) ρ + div (ρEu) Dt ∂t 12 Momentum equation in three dimensions • We will first derive conservation equations for momentum and energy for fluid particles. Next we will use the above relationships to transform those to an Eulerian frame (for fluid elements). • We start with deriving the momentum equations. • Newton’s second law: rate of change of momentum equals sum of forces. • Rate of increase of x-, y-, and z-momentum: ρ Du ρ Dv ρ Dw Dt Dt Dt • Forces on fluid particles are: – Surface forces such as pressure and viscous forces. – Body forces, which act on a volume, such as gravity, centrifugal, Coriolis, and electromagnetic forces. 13 Viscous stresses • Stresses are forces per area. Unit is N/m 2 or Pa. • Viscous stresses denoted by τ. τ • Suffix notation ij is used to indicate direction. • Nine stress components. τ τ τ – xx , yy , zz are normal stresses. τ E.g. zz is the stress in the z- direction on a z-plane. – Other stresses are shear τ stresses. E.g. zy is the stress in the y-direction on a z-plane. • Forces aligned with the direction of a coordinate axis are positive. Opposite direction is negative. 14 Forces in the x-direction ∂τ 1 τ + zx δ δ δ ( zx . z) y z ∂τ ∂τ ∂ yx 1 yx 1 z 2 − τ − δ δ δ τ + δ δ δ ( yx . y) x z ( yx . y) x z ∂ ∂y 2 y 2 ∂p 1 ∂p 1 ( p − . δx)δyδz − ( p + . δx)δyδz ∂x 2 ∂x 2 ∂τ xx 1 ∂τ − τ − δ δ δ xx 1 ( xx . x) y z (τ + . δx)δyδz ∂x 2 xx ∂x 2 z ∂τ 1 y − τ − zx δ δ δ ( zx . z) x y x ∂z 2 Net force in the x-direction is the sum of all the force components in that direction. 15 Momentum equation • Set the rate of change of x-momentum for a fluid particle Du/Dt equal to: – the sum of the forces due to surface stresses shown in the previous slide, plus – the body forces. These are usually lumped together into a source term S : M Du ∂(− p +τ ) ∂τ ∂τ ρ = xx + yx + zx + S Dt ∂x ∂y ∂z Mx τ – p is a compressive stress and xx is a tensile stress. • Similarly for y- and z-momentum: Dv ∂τ ∂(− p +τ ) ∂τ ρ = xy + yy + zy + S Dt ∂x ∂y ∂z My Dw ∂τ ∂τ ∂(− p +τ ) ρ = xz + yz + zz + S ∂ ∂ ∂ Mz Dt x y z 16 Energy equation • First law of thermodynamics: rate of change of energy of a fluid particle is equal to the rate of heat addition plus the rate of work done. • Rate of increase of energy is ρDE/Dt. • Energy E = i + ½ (u 2+v 2+w 2). • Here, i is the internal (thermal energy). • ½ (u 2+v 2+w 2) is the kinetic energy. • Potential energy (gravitation) is usually treated separately and included as a source term. • We will derive the energy equation by setting the total derivative equal to the change in energy as a result of work done by viscous stresses and the net heat conduction. • Next we will subtract the kinetic energy equation to arrive at a conservation equation for the internal energy. 17 Work done by surface stresses in x-direction ∂(uτ ) 1 (uτ + zx .