Simplification of Governing Equations (Different Types of Flow Models)

Depending on the flow conditions the Navier-Stokes equations, primarily the momentum equations, can be simplified. The simplification of these equations depends on which effects in the flow are significant or insignificant.

The simplified flow models that are most broadly used are the following:

Steady flow
Euler flow or Inviscid flow
Stokes flow
Incompressible flow

Steady Flow

The time dependence of flow field parameters is an important factor in the analysis of a fluid flow. A majority of flows are not steady but transient in nature. In a steady state flow the flow properties at a point, such as pressure and velocity, do not change with time. Steady state flows are of interest in cases where the flow properties need to be studied after the flow field has stabilized.

This is achieved in simulations by taking a very large time step

(Δ t)

which causes the time derivative of properties in the governing equations to reach to zero. (1)

\frac{\partial ρ}{\partial t} + \nabla \cdot (ρ \vec{u}) = 0

(2)

ρ \frac{\partial \vec{u}}{\partial t} + (ρ \vec{u} \cdot \nabla) \vec{u} = - \nabla p + ρ b + \nabla \cdot τ

(3)

ρ \frac{\partial h}{\partial t} + (ρ \vec{u} \cdot \nabla) h = \nabla \cdot (k \nabla T) + \nabla \vec{u} : τ + \frac{D p}{D t} + S

While performing simulations, a steady state flow result gives a preliminary insight into whether the problem is set up correctly or not. If there are a large number of oscillations in the residuals it can be inferred that the flow is transient and not steady. If the residuals show a smooth converge it implies that the flow field becomes stable and steady state is achieved. The results can also be used to fine tune the setup for further simulations or use them as initial conditions for a transient simulation.

Euler Flow or Inviscid Flow

Inviscid flow is a representation of a fluid flow where the dissipative and transport phenomenon of viscosity, mass diffusion and thermal diffusion are neglected. This assumption is valid when the viscous forces are small in comparison to the inertial forces.

Such flow situations can be identified in cases with a high value of Reynolds number, where the viscous effects are concentrated to regions close to solid boundary and can be neglected for the regions far away from the boundary.

The governing equations for such flows are expressed as: (4)

\frac{\partial ρ}{\partial t} + \nabla \cdot (ρ \vec{u}) = 0

(5)

ρ \frac{\partial \vec{u}}{\partial t} + (ρ \vec{u} \cdot \nabla) \vec{u} = - \nabla p + ρ b + \nabla \cdot τ

(6)

ρ \frac{\partial h}{\partial t} + (ρ \vec{u} \cdot \nabla) h = \nabla \cdot (k \nabla T) + \nabla \vec{u} : τ + \frac{D p}{D t} + S

These equations can form a closed form solution by assuming the equation of state.

Results obtained from these assumptions in the flow field are widely used in designing flying vehicles, rockets and their engines, turbines and compressors. Studies of inviscid flows are carried out in gas dynamics, acoustics, electro and magneto gas dynamics, the dynamics of rarefied gases, plasma dynamics, and so on.

Stokes Flow

Stokes flow is a representation of fluid flow where the viscosity of the fluid is high. In such flows the viscous effects dominate the advective inertial effects.

The Reynolds number in such flows is low, hence, it is also termed as creeping flow or Low Reynolds number flow.

The governing equations for such flows are expressed as: (7)

\frac{\partial ρ}{\partial t} + \nabla \cdot (ρ \vec{u}) = 0

(8)

ρ \frac{\partial \vec{u}}{\partial t} + (ρ \vec{u} \cdot \nabla) \vec{u} = - \nabla p + ρ b + \nabla \cdot τ

(9)

ρ \frac{\partial h}{\partial t} + (ρ \vec{u} \cdot \nabla) h = \nabla \cdot (k \nabla T) + \nabla \vec{u} : τ + \frac{D p}{D t} + S

This results in the linearization of the governing equations and thus they can be solved by a number of linear differential solvers available.

If the governing equations are non-dimensionalized and the Reynolds number is assumed to be very low, the momentum equation reduces to (10)

μ \nabla^{2} u = \nabla p - f

where (11)

f

is the external force.

Incompressible Flow

All fluids (gas or liquid) exhibit some change in volume when subjected to compressive stresses. The degree of compressibility for a fluid can be quantified using the bulk modulus of elasticity, E, defined as (12)

E = \frac{d p}{\frac{d ρ}{ρ}} o r \frac{d p}{- \frac{d V}{V}}

where $d p$ is the change in pressure and $d ρ$ or $d V$ is the corresponding change in volume or density.

A flow can be classified as incompressible if the density within fluid particle does not change during its motion. It is also termed as isochoric flow and implies that under certain conditions a compressible fluid can undergoincompressible flow. (13)

\frac{D ρ}{D t} = \frac{\partial ρ}{\partial t} + \vec{u} \cdot \nabla ρ = 0

Incompressibility is a property of flow and not of the fluid itself. Therefore the density field does not need to be uniform for the flow to be incompressible. When the above equation is combined with the continuity equation the following relation is obtained: (14)

\nabla \cdot \vec{u} = 0

which states that in an incompressible flow the velocity field is solenoidal (having zero divergence).

When an incompressibility assumption is made it is important to know under what conditions this assumption is valid. For a steady flow the condition is that the flow velocity must be much smaller compared to the local speed of sound, that is, $u ≪ a$ .

In case of an unsteady flow there is an additional condition which needs to satisfied, stated as $a t ≪ 1$ . Physically this condition states the distance travelled by a sound wave in time $t$ must be much greater than the distance travelled by the fluid particle. This implies that the propagation of pressure signals (sound waves) is instantaneous compared to the interval over which the flow field changes significantly.

When the limit for maximum relative change in density is set to five percent as the criteria for an incompressible flow, the maximum value of Mach number achieved is 0.3. This criteria states that any flow with a Mach number less than 0.3 (without heat source) can be assumed to be incompressible.

One of the implications of assuming the flow to be incompressible is that there is no equation of state as in a case of a compressible flow. In practice this means that the energy equation is decoupled from the continuity and momentum equations, assuming fluid properties are not a function of temperature. If the fluid properties change with temperature the equations again become coupled as in the case of a compressible flow.

The pressure in such a flow is no longer a thermodynamic quantity and cannot be related to temperature or density through an equation of state and must be obtained from the continuity and momentum equation while satisfying zero divergence for the velocity field. In the continuity equation there is no pressure term and in the momentum equation there are only the derivatives of pressure, but not the pressure itself. This means that the actual value of pressure in an incompressible flow solution is not important, only the changes of pressure in space are important.