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

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DNS of Multiphase Flows

Direct Numerical Simulations of Multiphase Flows-2
 


Governing Equations

Gretar Tryggvason DNS of Multiphase Flows Here we will focus on: Incompressible isothermal flow The “one-fluid” formulation of the governing equations DNS of Multiphase Flows

Governing Equations

DNS of Multiphase Flows The flow is predicted using the governing physical principles: Conservation of mass. If the density of a material particle does not change, we have incompressible flow 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 Control volume V Control surface S Conservation of mass ∂ ∂t Z

V

ρdv = I

S

ρu · nds The increase of mass inside a control volume is equal to the net inflow of mass (inflow minus

• utflow). The normal is

the outward pointing normal so inflow is negative and outflow is positive: Notice that the control volume may contain an interface separating fluids with different material properties, such as density. Normal vector n Interface, separating different fluids u DNS of Multiphase Flows The divergence (or Gauss’s) theorem can be used to convert surface integrals to volume integrals and vice versa. Applying it to the right hand side of the mass conservation equation gives

• r, bringing the time derivative under the integral and

collecting all terms under one integral sign Z I Z

V

r · udv = I

S

ρu · nds Z Z Z I ∂ ∂t Z

V

ρdv = Z

V

r · ρudv Z ⇣ ⌘ Z Z Z

V

⇣∂ρ ∂t + r · ρu ⌘ dv = 0

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DNS of Multiphase Flows The mass conservation equation equation is

Z

V

⇣∂ρ ∂t + r · ρu ⌘ dv = 0 Z

V

⇣∂ρ ∂t + u · rρ + ρr · u ⌘ dv = 0 Dρ ∂t = ∂ρ ∂t + u · rρ Z

V

⇣1 ρ Dρ Dt + r · u ⌘ dv = 0 Z

V

r · udv = 0 I

S

u · nds = 0

Expanding the divergence The first two terms are the convective derivative So we can write

if Dρ/Dt = 0

If then

• r

Volume is conserved! DNS of Multiphase Flows Conservation of momentum Stress Tensor Deformation Tensor

D = 1 2 ⇣ ru + uT ⌘

Di,j = 1 2 ✓ ∂ui ∂xj + ∂uj ∂xi ◆ Z I

The increase of momentum inside a control volume is equal to the net inflow of mass (inflow minus outflow) plus surface and volume forces Control volume V Control surface S Normal vector n u

• r, in component form:

∂ ∂t Z

V

ρudv = I

S

ρuu · nds I

S

nTds + Z

V

fdv

T = pI + 2µD Incompressible, Newtonian fluid DNS of Multiphase Flows The body force term generally includes gravity, but can also include other forces. Here, surface tension is treated as a body force so we write:

Z

V

fdv = Z

V

ρgdv + Z

V

fσdv I I

The evaluation of the gravity term is straightforward and how to find the surface tension is discussed below. DNS of Multiphase Flows Surface Tension

Fσ = (σt)2 (σt)1 t2 t1 = Z

δS

∂t ∂sds t2 t1 ∂t ∂s = κn Fσ = Z

δS

σ ∂t ∂sds = Z

δS

σκnds

The force on a control volume enclosing a segment of the interface is the difference in the tension where the interface enter the control volume and where it exists Notice that is the total force on the control volume due to the tension in the interface and

Fσ

Using Gives Where δS Assuming constant surface tension !. DNS of Multiphase Flows Fσ = Z

V

σκnδSds Multiply by a one dimensional delta function and integrate over the control volume to get the total surface force on the control volume The different ways in which we can write the surface force leads to different numerical approximations and in the numerical code we will write we will use a different form.

t2 t1Where δS

DNS of Multiphase Flows The material properties are functions of the marker function We are concerned with the flow of two or more fluids with different properties, such as density and viscosity. For immiscible fluids, the interface separating the different fluids remains sharp for all time. Identify each fluid by a marker function H

H(x) = ⇢ 1 in fluid 1 0 in fluid 2 h ⇣ ⌘i

ρ = Hρ1 + (1 H)ρ2 µ = Hµ1 + (1 H)µ2

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DNS of Multiphase Flows Gives If the density of each material particle remains constant, the material derivative of the density is zero

H(x) = ⇢ 1 in fluid 1 0 in fluid 2 h ⇣ ⌘i

∂H ∂t + u · rH = 0

ρ = Hρ1 + (1 H)ρ2 Dρ Dt = ∂ρ ∂t + u · rρ = 0

Substituting Substituting and using that ⌘ ✓ Dρ1 Dt = Dρ2 Dt = 0 In some cases we simply use the density as the indicator function DNS of Multiphase Flows

I

S

u · nds = 0 Z

Motion of the indicator function and updating properties Conservation of volume (from the mass conservation equation since the flow is incompressible) Momentum conservation (the Navier-Stokes equations)

V: Control volume S: Control surface

Summary: the governing equations in integral form.

∂H ∂t + u · rH = 0

• ∂

∂t Z

V

ρudv = I

S

ρuu · nds

• I

S

pnds + I

S

µ

• ru + (ru)T

· nds + Z

V

ρgdv + Z

V

σκnδSds

ρ = Hρ1 + (1 H)ρ2 µ = Hµ1 + (1 H)µ2

DNS of Multiphase Flows

• The conservation equations for mass and momentum

apply to any flow situation, including flows of multiple immiscible fluids.

• Each fluid generally has properties that are different from

the other constituents and the location of each fluid must therefore be tracked.

• We usually also have additional physics that must be

accounted for at the interface, such as surface tension.

• The governing equations can also be written in differential

form using using generalized functions DNS of Multiphase Flows

The “One-Fluid” Approach— The Governing Equations in Differential Form

DNS of Multiphase Flows A S H=1 H=0 Using: n s x y Generalized functions: Step function and delta function H(x, y, t) = Z

A(t)

δ(x x0)δ(y y0)da0 δ(x x0)δ(y y0) = δ(s)δ(n) ∂H ∂t + u · rH = 0 rH = Z

A

r[δ(x x0)δ(y y0)] da0 = Z

A

r0[δ(x x0)δ(y y0)] da0 = I

S

[δ(x x0)δ(y y0)n ds0 = Z

S

[δ(x x0)δ(y y0)n ds0 = Z

S

δ(s)δ(n)n ds0 = δ(n)n DNS of Multiphase Flows

Conservation of Momentum Conservation of Mass Equation of State: Singular interface term Incompressible flow Constant properties following a material point

The governing equations in differential form are derived in a standard way by consider the integral form for arbitrary control volume and insisting that the integrand is zero. In conservative form the equations are: Dρ Dt = 0; Dµ Dt = 0 r · u = 0 ∂ρu ∂t + r · ρuu = rp + r · µ(ru + rT u) + ρg + σκnfδ(n)

ρ = ρ1H + (1 H)ρ2 µ = µ1H + (1 H)µ2 DH Dt = ∂H ∂t + u · rH = 0

Alternatively, we have:

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DNS of Multiphase Flows The “one-fluid” formulation implicitly contains the proper interface jump conditions. Integrating each term over a small control volume centered on the interface:

=0 =0

The non-zero terms give the Jump Condition:

⇥ p ⇤ n ⇥ µ

• ru + uT ⇤

n κσn ⇥ ⇤ ⇥ ⇤ ⇥ p + µ

• ru + uT ⇤

= κσn

δV

Z

δV

Dρu Dt dv = Z

δV

rpdv + Z

δV

r · µ(ru + rT u)dv + Z

δV

ρgdv + Z

δV

κσnδ(n)dv

n

DNS of Multiphase Flows Write: and substitute into the momentum equation to get Interface conditions Momentum equation in phase 2 Momentum equation in phase 1 =0 =0 =0 We can also show that the “one-fluid” formulation contains the equations written separately for each fluid and the jump conditions: The terms multiplied by the different generalized functions must each vanish separately Use

u = H1u1 + H2u2 p = H1p1 + H2p2 ρ = H1ρ1 + H2ρ2

• H1H2 = 0

1 2

HiHi = Hi, i = 1, 2

H1(x)(. . . . . .) + H2(x)(. . . . . .) + δ(xf)(. . . . . .) = 0

DNS of Multiphase Flows

Solution Strategies

DNS of Multiphase Flows To solve for the flow, the governing equations are discretized both in space and time. The computational domain is divided into a finite number of control volumes (finite volume methods)

• r a finite number of points is used to represent the flow (finite

difference methods). For flows involving moving interfaces, solution methods can be divided in two major categories

• 1. Solving separate equations in each fluid using a moving

grid aligned with the interface, and applying boundary conditions at the interface:

• 2. Solving one set of equations for the whole domain on a

fixed grid and incorporate the boundary conditions into the equations DNS of Multiphase Flows Solving separate equations in each fluid using a moving grid aligned with the interface, and applying boundary conditions at the interface: Solving one set of equations for the whole domain on a fixed grid and incorporate the boundary conditions into the equations Stationary unstructured grid Stationary structured grid Body fitted structured grid Body fitted unstructured grid DNS of Multiphase Flows Here we solve one set of equations for the whole domain on a fixed grid and incorporate the boundary conditions into the equations The one-fluid formulation allows us to treat multi-phase flows in more or less the same way as single phase flows. The main differences are: The density and viscosity change discontinuously across the interface and have to be updated as the interface moves Surface tension needs to be evaluated and added to the Navier-Stokes equations