Governing not change, we have incompressible flow Equations - PDF document


 DNS of Multiphase Flows DNS of Multiphase Flows Direct Numerical Simulations of Here we will focus on: Multiphase Incompressible isothermal flow Flows-2 
 The “one-fluid” formulation of the governing equations Governing Equations Gretar Tryggvason DNS of Multiphase Flows DNS of Multiphase Flows The flow is predicted using the governing physical principles: Conservation of mass. If the density of a material particle does Governing not change, we have incompressible flow Equations Conservation of momentum. For incompressible flow the pressure is adjusted to enforce conservation of volume Conservation of energy. For isothermal flow as we will be concerned with here, the energy equation is not needed Geometric relationships that specify the motion of fluid particles. For flow consisting of two or more fluids where each fluid has constant properties, we only need to know how the interface moves DNS of Multiphase Flows DNS of Multiphase Flows Normal Conservation of mass The divergence (or Gauss’s) theorem can be used to Z I vector n Control convert surface integrals to volume integrals and vice volume V The increase of mass versa. inside a control volume is Z I Control r · u dv = ρ u · n ds equal to the net inflow of surface S V S mass (inflow minus Z Z outflow). The normal is Applying it to the right hand side of the mass Z I Interface, the outward pointing conservation equation gives u separating normal so inflow is different fluids negative and outflow is ∂ Z Z ρ dv = � r · ρ u dv positive: ∂ t V V Notice that the control or, bringing the time derivative under the integral and Z ⇣ Z Z ⌘ volume may contain an collecting all terms under one integral sign ∂ Z I ρ dv = � ρ u · n ds interface separating fluids ∂ t Z ⇣ ∂ρ V S with different material ⌘ ∂ t + r · ρ u dv = 0 properties, such as density. V

DNS of Multiphase Flows DNS of Multiphase Flows Normal Conservation of momentum The mass conservation equation equation is vector n Control The increase of momentum Z ⇣ ∂ρ ⌘ volume V ∂ t + r · ρ u dv = 0 inside a control volume is V equal to the net inflow of Control Expanding the divergence mass (inflow minus outflow) surface S Z ⇣ ∂ρ ⌘ plus surface and volume ∂ t + u · r ρ + ρ r · u dv = 0 V forces The first two terms are the convective derivative u ∂ Z I I Z ρ u dv = � ρ uu · n ds � nT ds + f dv D ρ ∂ t = ∂ρ ∂ t + u · r ρ ∂ t V S S V Stress Tensor So we can write Incompressible, Newtonian fluid T = � p I + 2 µ D Z ⇣ 1 D ρ ⌘ Dt + r · u dv = 0 ρ V Deformation Tensor ✓ ∂ u i Volume is I D = 1 D i,j = 1 + ∂ u j ◆ Z ⇣ r u + u T ⌘ or, in component form: or u · n ds = 0 If if D ρ /Dt = 0 then r · u dv = 0 conserved! 2 2 ∂ x j ∂ x i S V Z I DNS of Multiphase Flows DNS of Multiphase Flows Surface Tension The body force term generally includes gravity, but can The force on a control volume enclosing a also include other forces. Here, surface tension is Where δ S segment of the interface is the difference � t 1 treated as a body force so we write: in the tension where the interface enter t 2 the control volume and where it exists F σ = ( σ t ) 2 � ( σ t ) 1 Z Z Z f dv = ρ g dv + f σ dv Using Assuming V V V Z ∂ t constant ∂ t I I t 2 � t 1 = ∂ sds and ∂ s = κ n surface δ S The evaluation of the gravity term is straightforward and tension ! . Gives how to find the surface tension is discussed below. Z Z σ ∂ t F σ = ∂ sds = σκ n ds δ S δ S Notice that is the total force on the control F σ volume due to the tension in the interface DNS of Multiphase Flows DNS of Multiphase Flows We are concerned with the flow of two or more fluids with Multiply by a one dimensional delta function and integrate over the different properties, such as density and viscosity. � t 1 Where δ S control volume to get the total surface For immiscible fluids, the interface separating the different force on the control volume t 2 fluids remains sharp for all time. Identify each fluid by a marker function H Z ⇢ 1 in fluid 1 F σ = σκ n δ S ds V H ( x ) = 0 in fluid 2 The material properties are h ⇣ ⌘i The different ways in which we can write the surface force functions of the marker function leads to different numerical approximations and in the ρ = H ρ 1 + (1 � H ) ρ 2 numerical code we will write we will use a different form. µ = Hµ 1 + (1 � H ) µ 2

DNS of Multiphase Flows DNS of Multiphase Flows If the density of each material Summary: the governing equations in integral form. particle remains constant, the � � Momentum conservation (the Navier-Stokes equations) material derivative of the density is zero ∂ Z I ρ u dv = � ρ uu · n ds ∂ t D ρ Dt = ∂ρ V S ∂ t + u · r ρ = 0 I I Z Z r u + ( r u ) T � p n ds + µ � · n ds + ρ g dv + σκ n δ S ds � S S V V Substituting Conservation of volume (from the mass conservation ✓ ⇢ ρ = H ρ 1 + (1 � H ) ρ 2 1 in fluid 1 ⌘ equation since the flow is incompressible) H ( x ) = 0 in fluid 2 Substituting and using that I u · n ds = 0 D ρ 1 Dt = D ρ 2 Dt = 0 h ⇣ ⌘i S Z Motion of the indicator function and updating properties Gives In some cases we simply ∂ H ρ = H ρ 1 + (1 � H ) ρ 2 use the density as the ∂ H ∂ t + u · r H = 0 ∂ t + u · r H = 0 V : Control volume µ = Hµ 1 + (1 � H ) µ 2 indicator function S : Control surface DNS of Multiphase Flows DNS of Multiphase Flows • The conservation equations for mass and momentum The “One-Fluid” apply to any flow situation, including flows of multiple immiscible fluids. Approach — • Each fluid generally has properties that are different from the other constituents and the location of each fluid must therefore be tracked. The Governing • We usually also have additional physics that must be Equations in accounted for at the interface, such as surface tension. • The governing equations can also be written in differential Differential Form form using using generalized functions DNS of Multiphase Flows DNS of Multiphase Flows The governing equations in differential form are derived in a Generalized functions: Step function and delta function s standard way by consider the integral form for arbitrary control n Z H ( x, y, t ) = δ ( x � x 0 ) δ ( y � y 0 ) da 0 volume and insisting that the integrand is zero. In conservative A ( t ) form the equations are: A Z r [ δ ( x � x 0 ) δ ( y � y 0 )] da 0 r H = Conservation of Momentum H= 1 A H=0 y ∂ρ u Z ∂ t + r · ρ uu = �r p + r · µ ( r u + r T u ) + ρ g + σκ n f δ ( n ) r 0 [ δ ( x � x 0 ) δ ( y � y 0 )] da 0 = � S A Singular I x interface term [ δ ( x � x 0 ) δ ( y � y 0 ) n ds 0 = � Conservation of Mass S ∂ H ∂ t + u · r H = 0 Z r · u = 0 Incompressible flow Alternatively, we have: [ δ ( x � x 0 ) δ ( y � y 0 ) n ds 0 = � DH Dt = ∂ H S ∂ t + u · r H = 0 Equation of State: Constant Z δ ( s ) δ ( n ) n ds 0 = � properties Using: D ρ Dµ ρ = ρ 1 H + (1 � H ) ρ 2 Dt = 0; Dt = 0 S following a µ = µ 1 H + (1 � H ) µ 2 δ ( x � x 0 ) δ ( y � y 0 ) = δ ( s ) δ ( n ) = � δ ( n ) n material point