# Non-dimensionalization and scaling of the Navier–Stokes equations

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In fluid mechanics, non-dimensionalization of Navier–Stokes equation is the conversion of the Navier–Stokes equation to a form which is easier to use and reducing the number of free parameters in the problem to be studied. The non-dimensionalized Navier–Stokes equation is beneficial to use when posed with similar physical situations that is problems where the only change are in the basic dimensions of the system. Scaling of Navier–Stokes equation refers to the process of selecting the proper scales for non-dimensionalization of the equation. Since the final equation needs to be dimensionless, using the parameters and constants of the equation a suitable combination of these is found having the same dimensions as that of the variables in the equation. As a result of this combination the number of parameters to be analyzed is reduced and the results are obtained in terms of scaled variables.

## Need for non-dimensionalization and scaling

In addition to reducing the number of parameters, non-dimensionalized equation helps to gain a greater insight into the relative size of various terms present in the equation.[1][2] Following appropriate selecting of scales for the non-dimensionalization process, this leads to identification of small terms in the equation. Neglecting the smaller terms against the bigger ones allows for the simplification of the situation. For the case of flow without heat transfer, the non-dimensionalized Navier–Stokes equation depend only on the Reynolds Number and hence all physical realizations of the related experiment will have the same value of non-dimensionalized variables for the same Reynolds Number.[3]

Scaling helps provide better understanding of the physical situation, with the variation in dimensions of the parameters involved in the equation. This allows for experiments to be conducted on smaller scale prototypes provided that any physical effects which are not included in the non-dimensionalized equation are unimportant.

## Non-dimensionalized Navier–Stokes equation

The Navier–Stokes equation is written as:

$\rho \left(\frac{\partial \mathbf{u}}{\partial t} + \mathbf{u} \cdot \nabla \mathbf{u}\right) = -\nabla p + \mu \nabla^2 \mathbf{u} + \mathbf{f}.$ [4][5]

where ρ is the density of fluid, P is the pressure, μ is the viscosity and f is the body force.

The above equation can be dimensionalized through selection of appropriate scales as follows:

Scale dimensionless variable
Length L $x^*\ = \frac{x}{L}\,, \ y^*\ = \frac{y}{L} \,, z^*\ = \frac{z}{L}\,$
Velocity U $u^*\ = \frac{u}{U}\,$
Time L/U $t^*\ = \frac{t}{L/U}\,$
Pressure: there is no natural selection for the pressure scale. Where dynamic effects are dominant i.e. high velocity flows
$p^* = \frac{p}{\rho U^2}$

Where viscous effects are dominant i.e. creeping flows

$p^* = \frac{p L}{\mu U}$

Substituting the scales the non-dimensionalized equation obtained is:

$\frac{\partial \mathbf{u^*}}{\partial t^*} + \mathbf{u^*} \cdot \nabla \mathbf{u^*}\ = -\nabla p^* + \frac{1}{Re} \nabla^2 \mathbf{u^*} + \mathbf{f}\frac{L}{\rho U^2}.$[4]

(1)

### Gravitational force is the only body force

If the gravitational force is the only body force then the term gL/U2 can be substituted with 1/Fr, where Fr is the Froude number defined as

$Fr = \frac{U^2}{gL}.$

This is only valid if there is a free surface.

### Flows with large viscosity

For flows where viscous forces are dominant i.e. slow flows with large viscosity, a viscous pressure scale μU/L is used. In the absence of a free surface, the equation obtained is

$Re \left( \frac{\partial \mathbf{u^*}}{\partial t^*} + \mathbf{u^*} \cdot \nabla \mathbf{u^*} \right)\ = -\nabla p^* + \nabla^2 \mathbf{u^*}.$

(2)

### When Re → 0

Scaling of equation (1) can be done, in a flow where inertia term is smaller than the viscous term i.e. when Re → 0 then inertia terms can be neglected, leaving the equation of a creeping motion.

$\frac{\partial \mathbf{u^*}}{\partial t} = -\nabla p^* + \frac{1}{Re} \nabla^2 \mathbf{u^*}.$

Such flows tend to have influence of viscous interaction over large distances from an object.[citation needed] At low pressures the same equation reduces to a diffusion equation

$\frac{\partial \mathbf{u^*}}{\partial t} = \frac{1}{Re} \nabla^2 \mathbf{u^*}.$

### When Re → ∞

Similarly if Re → ∞ i.e. when the inertia forces dominates, the viscous force can be neglected by substituting μ = 0 in the equation.The non-dimensionalized Euler equation for an inviscid flow is

$\frac{\partial \mathbf{u^*}}{\partial t} + \mathbf{u^*} \cdot \nabla \mathbf{u^*}\ = -\nabla p^*.$[6]

### When density varies due to both concentration and temperature

Density variation due to both concentration and temperature is an important field of study in Double diffusive convection"If density changes due to both temperature and salinity is taken into account then some more terms add to the Z-Component of momentum as follows:[7] [8]

$\frac{\partial W}{\partial t} + U \frac{\partial W}{\partial X} + W \frac{\partial W}{\partial Z}\ = -\frac{1}{\rho_o}\frac{\partial p_d}{\partial Z} + v \left(\frac{\partial^2 W}{\partial X^2} + \frac{\partial^2 W}{\partial Z^2}\right)\ - g \left(\beta_{s}\nabla{S} - \beta_{T}\nabla{T}\right)$

Where S is the salinity of the fluid, βT is the thermal expansion coefficient at constant pressure and βS is the coefficient of saline expansion at constant pressure and temperature.

Non dimensionalizing using the scale:

$S^* = \frac{S - S_B}{S_T - S_B}$ and $T^* = \frac{T - T_B}{T_T - T_B}$

we get

$\frac{\partial W^*}{\partial t^*} + U^* \frac{\partial W^*}{\partial X^*} + W^* \frac{\partial W^*}{\partial Z^*}\ = -\frac{\partial p_d}{\partial Z^*} + Pr \left(\frac{\partial^2 W^*}{\partial X^{*2}} + \frac{\partial^2 W^*}{\partial Z^{*2}}\right)\ - {Ra_s Pr_s S} + {Ra_T Pr_T T}$

where ST, TT denote the salinity and temperature at top layer, SB, TB denote the salinity and temperature at bottom layer, Ra is the Rayleigh Number, and Pr is the Prandtl Number. The sign of RaS and RaT will change depending on whether it stabilizes or destabilizes the system.

## References

1. ^ Versteeg H.K, An introduction to computational fluid dynamics: the finite volume method, 2007, prentice hall, 9780131274983
2. ^ Patankar Suhas V. , Numerical heat transfer and fluid flow, 1980, Taylor & Francis, 9780891165224
3. ^ Salvi Rodolfo, The Navier Stokes equation theory and numerical methods, 2002, M. Dekker, 9780824706722
4. ^ a b Fox, Robert W.; Alan T. McDonald; Philip J. Pritchard (2006). Introduction to fluid mechanics (6th ed.). Hoboken, NJ: Wiley. p. 213–215. ISBN 9780471735588.
5. ^ Tritton, D.J. (1988). Physical fluid dynamics (2nd ed.). Oxford [England]: Clarendon Press. pp. 55–58. ISBN 0198544898.
6. ^ White, Frank M. (2003). Fluid mechanics (5th ed.). Boston: McGraw-Hill. pp. 188–189. ISBN 9780072402179.
7. ^ On the relationship between ﬁnger width, velocity, and ﬂuxes in thermohaline convection, 2009, K. R. Sreenivas, O. P. Singh & J. Srinivasan, Phys. Fluids (American Institute of Physics) 21(2), pp. 026601.
8. ^ Approximation of the hydrostatic Navier-Stokes system for density stratiﬁed ﬂows by a multilayer model. Kinetic interpretation and numerical validation, E. Audusse a,b , M.-O. Bristeau , M. Pelanti , J. Sainte-Marie, aUniversité Paris 13, Institut Galilée, 99 avenue Jean-Baptiste Clément, 93430 Villetaneuse, France. b INRIA Rocquencourt, B.P. 105, 78153 Le Chesnay Cedex, France. c Saint-Venant Laboratory, 6 quai Watier, 78400 Chatou, France.
• "Non-dimensionalizing Navier–Stokes". CFD Online. Retrieved 11 October 2012.
• T.Cebeci J.RShao,F. Kafyeke E. Laurendeau, Computational Fluid Dynamics for Engineers, Springer, 2005
• C. Pozrikidis, FLUID DYNAMICS Theory, Computation, and Numerical Simulation, KLUWER ACADEMIC PUBLISHERS, 2001

## Further reading

• Tritton, D.J. (1988). "Chapter 7 – Dynamic similarity". Physical fluid dynamics (2nd ed.). Oxford [England]: Clarendon Press. ISBN 0198544898.
• Mattheij, R.M.M.; Rienstra, S.W.; ten Thije Boonkkamp, J.H.M. (2005). "§7.4 – Scaling and Reduction of the Navier–Stokes Equations". Partial Differential Equations: Modeling, Analysis, Computation. SIAM. pp. 148–155. ISBN 9780898715941.
• Graebel, William (2007). "§6.2 – The Boundary Layer Equations". Advanced Fluid Mechanics. Academic Press. pp. 171–174. ISBN 9780123708854.
• Leal, L. Gary (2007). Advanced Transport Phenomena: Fluid Mechanics and Convective Transport Processes. Cambridge University Press. ISBN 9780521849104.
This book contains several examples of different non-dimensionalizations and scalings of the Navier–Stokes equations
• Krantz, William B. (2007). Scaling Analysis in Modeling Transport and Reaction Processes: A Systematic Approach to Model Building and the Art of Approximation. John Wiley & Sons. ISBN 9780471772613.
• Zeytounian, Radyadour Kh. (2002). Asymptotic Modelling of Fluid Flow Phenomena. Fluid Mechanics and Its Applications 64. Kluwer Academic Publishers. ISBN 978-1-4020-0432-2.