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The Dimensionless Navier-Stokes Equation

Key Takeaways

  • Scaling and normalization are two methods used to reduce differential equations into a non-dimensional form.

  • Non-dimensionalized differential equations and their solutions can be generalized to nearly any system.

  • In fluid dynamics, the dimensionless Navier-Stokes equation is the main tool used to generalize fluid flow behavior into new domains and systems.

Fluid dynamics equations

A standard method used in the analysis of differential equations is normalization, which redefines the solution, the governing space/time variables, or both as dimensionless quantities. Normalization and non-dimensionalization are effectively the same process when applied to the space and time variables that are defined in a differential equation. In fields like structural analysis, fluid dynamics, and electromagnetics, normalization is an important technique that aids scaling so that a solution can be adapted into a new geometry or system.

How is scaling performed and what does it say about fluid flow in various systems? To see how this is applied and generalized to compressible viscid flows, we can look to the dimensionless Navier-Stokes equation of motion for the fluid flow field. This equation of motion may not have an obvious scaling distance in some cases, but the creative selection of the relevant normalization constant can make certain flow problems easier to analyze and solve.

Deriving a Dimensionless Navier-Stokes Equation

The Navier-Stokes equation is not normally presented in a dimensionless form. Instead, it is based on some absolute unit system (metric or imperial) that is used to define length scales, time scales, pressure scales, or flow velocity scales. These “scale” terms refer to normalization constants, although unlike the case in typical analyses of differential equations, these constants may not correspond to specific dimensional or material aspects of the system.

To start, we should note that fluid dynamics discussions start with the Navier-Stokes equation, which is an equation of motion for a flowing fluid that describes the conservation of the fluid’s momentum. When we consider the sum of internal and external forces acting on a fluid, we can arrive at the following equation of motion describing the fluid flow:

Navier-Stokes equation

Navier-Stokes momentum equation for compressible flows

This is among the most generalized forms of the Navier-Stokes equation as written in vector form. The form shown includes non-zero viscosity as well as a linear stress strain relation relating compressibility and fluid density as defined in the proportionality constant λ. The above equation, however, is still in a dimensional form, meaning the various terms in the expression (fluid flow rate u, density ⍴, etc.) are defined in terms of some measurement units. If we want to scale the above equation, one method that helps you get started is to define a dimensionless equivalent of the above equation using a relevant length, time, fluid flow rate, or pressure scale. Let’s look at how to do that.

Defining Scales

To define a dimensionless Navier-Stokes equation, we need to determine some possible normalization scales that can be applied to the various terms in the above equation. Within the above equation, one can see four possible normalization scales that can be used to define a dimensionless Navier-Stokes equation. These are used to normalize the space, time, pressure, or flow variable; they also modify the gradient operator such that it defines a rate of change with respect to a dimensionless normalized variable.

Variable

Normalized Variable

Normalized Gradient Operator

Spatial coordinate r

r* = r/L

∇* = L

Flow rate u

u* = u/U

N/A

Time coordinate t

t* = t/(L/U)

N/A

Pressure p

High viscosity: p* = (pL)/(μU)

 

High velocity: p* = p/(⍴U2)

N/A

The normalization constant for the spatial variable and the time variable is arbitrary. For the spatial variable, the normalization constant could be some characteristic length scale in the system, such as the distance between two boundaries or the size of some particular feature in the system. A similar consideration can be made for the flow rate, such as the free-stream flow rate. 

For the time variable, we could also apply a totally arbitrary normalization constant instead of using the flow rate and length scale as the basis for normalization. This would then enforce some requirement on the flow rate for a given length scale or vice versa. The resulting solution to the dimensionless Navier-Stokes equation will be invariant under any additional scale transformation in time because the Navier-Stokes equation is linear in time (assuming p is static or also linear in time).

Applying the Linear Transformation

The linear transformations defined above can now be applied to the Navier-Stokes equation of motion to produce the following non-dimensional result:

 Dimensionless Navier-Stokes equation

Dimensionless Navier-Stokes equation

Note that the result is defined in terms of two dimensionless quantities that are important in fluid dynamics: the Reynolds number (Re) and the Froude number (Fr). We can define similar equations for the fluid dissipation and enthalpy by applying the transformations shown above to these additional equations. This reduces the full Navier-Stokes equations to their dimensionless form in terms of normalized variables.

Using the Dimensionless Navier-Stokes Equation in a Solver

The Navier-Stokes equation in its dimensionless form is just as complex as in its dimensionalized form, but the benefit arises when we consider the system behavior in terms of the relevant length, time, pressure, or flow rate scales in the system. CFD simulation packages are excellent tools for solving the dimensionless Navier-Stokes equation, as the solution can provide significant insight in any similar system that is scaled in length, time, or pressure.

The complete set of fluid dynamics analysis and simulation tools in Omnis 3D Solver from Cadence are ideal for defining and running CFD simulations with modern numerical approaches. This includes solving the dimensionless Navier-Stokes equations in an arbitrary system.

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