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Formulating the 2D Incompressible Steady-State Navier-Stokes Equation

Key Takeaways

  • The types of fluid flow regimes are steady-state flow, pseudo steady-state flow, and transient or unsteady-state flow.

  • Steady-state flow refers to conditions where fluid properties such as temperature, pressure, and velocity do not change over time.

  • There are two momentum equations corresponding to the velocity components in the x and y directions of the 2D flow.

Navier-Stokes equation

Navier-Stokes equations can be applied to solve fluid flow problems involving different flow regimes

In fluid flow problems, Navier-Stokes equations are used to develop numerical models. These equations apply to compressible and incompressible fluid flows. According to the flow characteristics, the fluid flow can be subclassified into 1-D, 2-D, 3-D, steady-state, or transient state. It is possible to apply the Navier-Stokes equations to solve different flow regimes. For example, you can solve the 2-D incompressible steady-state Navier-Stokes equation, and similarly, you can get numerical solutions for the 3-D compressible unsteady-state flow. In this article, we will discuss flow regimes and the 2-D incompressible steady-state Navier-Stokes equation.

Types of Flow Regimes in Fluid Dynamics

Fluid flow can be classified based on the rate of change of pressure with respect to time. Flow types are:

  1. Steady-State
    If the pressure of the fluid remains constant over time, such a type of fluid flow is called steady-state flow. The steady-state flow can be mathematically expressed as:

Steady state flow

  1. Pseudo Steady-State
    When the pressure drop is a constant for each unit of time, the fluid flow is called pseudo steady-state. Mathematically, pseudo steady-state can be represented as:

Pseudo steady state

  1. Transient or Unsteady-State
    When the fluid pressure variation is a function of fluid properties or fluid domain geometry, then such fluid flow forms a transient state or unsteady-state flow. 

Unsteady state flow

2D Incompressible Steady-State Flow

In this section, we will break each word in “2D incompressible steady-state fluid flow” down and see what is the charactersistic of the flow.

Generally, steady-state flow refers to the condition where fluid properties such as temperature, pressure, and velocity do not change over time. With a change of temperature or pressure, the volume of real fluids change. The fluids whose volume changes is called compressible fluids and the property is called compressibility.

Most fluids are compressible in nature. However, to simplify the flow analysis, small changes in fluid pressure are neglected and the flow is assumed to be incompressible. In most cases, compressible liquids are treated as incompressible. Incompressible fluid cannot be compressed or expanded and the volume remains constant.

We have learned what an incompressible steady-state flow is, now let’s find out more details about 2D fluid flow. When fluid properties such as pressure and velocity vary in three dimensions, flow is considered to be 3D flow. However, the greatest variations in fluid properties never occur in all directions; most fluid properties change only in one or two directions. When flow parameters such as velocity and pressure at a given time only vary in the flow direction and not across the cross-section, flow is 1D. When the flow parameters vary in the flow direction and in the direction perpendicular to the flow, flow is considered 2D. For example, the velocity of the flow vary in two directions of x and y. 

The Navier-Stokes Equations

In fluid dynamics, changes in properties such as speed, pressure, and density can be addressed using the Navier-Stokes equations. The Navier-Stokes equations describe the conservation of mass, momentum, and energy and model fluid flow problems numerically. The Navier-Stokes equations are a collection of three fundamental equations:

  1. Continuity equation, which expresses the conservation of mass:

Continuity equation

  1. Newton’s second law, which expresses the conservation of momentum:

Newton’s second law

  1. The first law of thermodynamics, which expresses the conservation of energy:

Conservation of energy

2D Incompressible Steady-State Navier-Stokes Equation

In incompressible steady-state flow, there is no linking between density and pressure. The mass conservation is a constraint on the velocity field. The velocity V is a two-dimensional vector represented as V = (u, v); u and v are the velocity components along the x and y axis. The first equation (continuity equation) of Navier-Stokes equations represents the incompressibility property of the fluid. As the fluid flow is in a steady-state, the field properties are not functions of time and the equation reduces to one comprising of velocity vector.

The continuity equation of 2D incompressible steady-state flow in a differential form can be written as:

Continuity equation

The 2D Navier–Stokes equations explain the momentum conservation of incompressible fluid. There are two momentum equations corresponding to the velocity components in the x and y directions of 2D flow. The governing equations corresponding to the conservation of momentum for 2D incompressible steady-state flow in differential form can be written as:

X-momentum equation:

 X momentum equation

Y-momentum equation:

Y momentum equation

Note that p is static pressure, is the  density, and is the dynamic viscosity of the flowing fluid.

The momentum equations in 2D incompressible steady-state Navier-Stokes equations can be expressed in dimensionless form using the Reynolds number. To solve 2D incompressible steady-state Navier-Stokes equations, approximations and assumptions are made so that the appropriate solutions can be achieved. Numerical techniques such as the finite volume, finite element, and finite difference methods can be used to solve the Navier-Stokes equations.

Cadence’s CFD simulation tools can be used to solve the Navier-Stokes equations in fluid mechanics problems. Cadence’s suite of CFD software can run CFD simulations to solve these equations for any flow regime.

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