View source for Non-dimensionalization and scaling of the Navier–Stokes equations

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{{main|Navier–Stokes equations|Non-dimensionalization|Scale invariance}}
{{continuum mechanics}}

In [[fluid mechanics]], '''non-dimensionalization of the Navier–Stokes equations''' is the conversion of the [[Navier–Stokes equation]] to a [[nondimensionalization|nondimensional form]]. This technique can ease the analysis of the problem at hand, and reduce the number of [[free parameter]]s. Small or large sizes of certain dimensionless parameters indicate the importance of certain terms in the equations for the studied flow. This may provide possibilities to neglect terms in (certain areas of) the considered flow. Further, non-dimensionalized Navier–Stokes equations can be beneficial if one is posed with similar physical situations – that is problems where the only changes are those of the basic [[dimensional analysis|dimension]]s of the system.

Scaling of Navier–Stokes equation refers to the process of selecting the proper [[spatial scale]]s – for a certain type of flow – to be used in the non-dimensionalization of the equation. Since the resulting equations need to be dimensionless, a suitable combination of parameters and constants of the equations and flow (domain) characteristics have to be found. As a result of this combination, the number of parameters to be analyzed is reduced and the results may be obtained [[scale invariance|in terms of the 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.<ref>Versteeg H.K, An introduction to computational fluid dynamics: the finite volume method, 2007, prentice hall, 9780131274983</ref><ref>Patankar Suhas V., Numerical heat transfer and fluid flow, 1980, Taylor & Francis, 9780891165224</ref>
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 depends 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.<ref>Salvi Rodolfo, The Navier Stokes equation theory and numerical methods, 2002, M. Dekker, 9780824706722</ref>

[[scale invariance|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 incompressible Navier–Stokes momentum equation is written as:

:<math>\frac{\partial \mathbf{u}}{\partial t} + (\mathbf{u} \cdot \nabla) \mathbf{u} = -\frac 1 \rho \nabla p + \nu \nabla^2 \mathbf{u} + \mathbf{g}.</math> <ref name=Robert/><ref>{{cite book|last=Tritton|first=D.J. |author1-link=David Tritton |pages=55&ndash;58 |title=Physical fluid dynamics|year=1988|publisher=Clarendon Press|location=Oxford [England]|isbn=0198544898|edition=2nd}}</ref>

where ρ is the [[density]], ''p'' is the [[pressure]], &nu; is the [[kinematic viscosity]], '''u''' is the [[flow velocity]], and '''g''' is the body acceleration field.

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

:{| class="wikitable"
|-
! scope="col" style="width:200px;"| Scale 
! scope="col" style="width:200px;"| dimensionless variable
|-
| [[Length]] ''L''
| <math>\mathbf r^*\ = \frac{\mathbf r}{L}</math> and <math>\nabla^*\ = L\nabla</math>
|-
| [[Flow velocity]] ''U'' 
| <math>\mathbf u^*\ = \frac{\mathbf u}{U}\,</math>
|-
| [[Time]] ''L''/''U''
| <math>t^*\ = \frac{t}{L/U}\,</math>
|-
| [[Pressure]]: there is no natural selection for the pressure scale.
| Where dynamic effects are dominant i.e. high velocity flows
:<math> p^* = \frac{p}{\rho U^2} </math>
Where viscous effects are dominant i.e. creeping flows
:<math> p^* = \frac{p L}{\mu U} </math>
|}

Substituting the scales the non-dimensionalized equation obtained is:

{{NumBlk|:|<math>
\frac{\partial \mathbf{u^*}}{\partial t^*} + (\mathbf{u^*} \cdot \nabla^*) \mathbf{u^*}\ = -\nabla^* p^* + \frac{1}{Re} \nabla^{*2} \mathbf{u^*} +  \frac{1}{Fr^2} \hat{g}.</math><ref name =Robert>{{cite book|last=Fox|first=Robert W.|author2=Alan T. McDonald |author3=Philip J. Pritchard  |title=Introduction to fluid mechanics|url=https://archive.org/details/introductiontofl00foxr_614|url-access=limited|year=2006|publisher=Wiley|location=Hoboken, NJ|isbn=9780471735588|edition=6th|page=[https://archive.org/details/introductiontofl00foxr_614/page/n226 213]&ndash;215}}</ref>
|{{EquationRef|1}}}}

where <math> Fr </math> is the [[Froude number]] and <math> Re </math> is the [[Reynolds number]] (<math> Re = UL/\nu </math>).

=== Flows with large viscosity ===
For flows where [[Viscosity|viscous force]]s are dominant i.e. slow flows with large viscosity, a viscous pressure scale μ''U''/''L''  is used. In the absence of a free surface, the equation obtained is

{{NumBlk|:|<math>
Re \left( \frac{\partial \mathbf{u^*}}{\partial t^*} + (\mathbf{u^*} \cdot \nabla^* )\mathbf{u^*} \right)\ = -\nabla^* p^* + \nabla^{*2} \mathbf{u^*}.</math>
|{{EquationRef|2}}}}

=== Stokes regime ===
{{see also|Stokes flow}}
Scaling of equation ({{EquationNote|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]].

:<math>Re \frac{\partial \mathbf{u^*}}{\partial t^*} = -\nabla^* p^* + \nabla^{*2} \mathbf{u^*}.</math>

Such flows tend to have influence of viscous interaction over large distances from an object.{{citation needed|date=September 2012}} At low Reynolds number the same equation reduces to a [[diffusion equation]], named '''Stokes equation'''

:<math> -\nabla^* p^* + \nabla^{*2} \mathbf{u^*}= \mathbf 0.</math>

=== Euler regime ===
Similarly if Re → ∞ i.e. when the inertia forces dominates, the viscous contribution can be neglected. The non-dimensionalized [[Euler equations (fluid dynamics)|Euler equation]] for an [[inviscid flow]] is

:<math>\frac{\partial \mathbf{u^*}}{\partial t} + (\mathbf{u^*} \cdot \nabla^* )\mathbf{u^*}\ = -\nabla^* p^*.</math><ref>{{cite book|last=White|first=Frank M.|title=Fluid mechanics|url=https://archive.org/details/fluidmechanics00whit_184|url-access=limited|year=2003|publisher=McGraw-Hill|location=Boston|isbn=9780072402179|edition=5th|pages=[https://archive.org/details/fluidmechanics00whit_184/page/n198 188]–189}}</ref>

=== 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 are taken into account, then some more terms add to the Z-Component of momentum as follows:<ref>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.</ref>
<ref>Approximation of the hydrostatic Navier–Stokes system for density stratified flows by a multilayer model. Kinetic interpretation and numerical validation, E. Audusse, M.-O. Bristeau, M. Pelanti, J. Sainte-Marie, Université Paris 13, Institut Galilée, 99 avenue Jean-Baptiste Clément, 93430 Villetaneuse, France. INRIA Rocquencourt, B.P. 105, 78153 Le Chesnay Cedex, France. Saint-Venant Laboratory, 6 quai Watier, 78400 Chatou, France.</ref>
:<math> \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)</math>

Where ''S'' is the salinity of the fluid, β<sub>''T''</sub> is the thermal expansion coefficient at constant pressure and β<sub>''S''</sub> is the coefficient of saline expansion at constant pressure and temperature.

Non dimensionalizing using the scale:
:<math> S^* = \frac{S - S_B}{S_T - S_B}</math> and <math> T^* = \frac{T - T_B}{T_T - T_B}</math>
we get

:<math> \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}</math>

where ''S<sub>T</sub>'', ''T<sub>T</sub>'' denote the salinity and temperature at top layer, ''S<sub>B</sub>'', ''T<sub>B</sub>'' denote the salinity and temperature at bottom layer, Ra is the [[Rayleigh Number]], and Pr is the [[Prandtl Number]]. The sign of Ra<sub>S</sub> and Ra<sub>T</sub> will change depending on whether it stabilizes or destabilizes the system.

==References==
===Footnotes===
{{Reflist}}

===Other===
{{Refbegin}}
*{{cite web|title=Non-dimensionalizing Navier–Stokes|url=http://www.cfd-online.com/Forums/main/94617-non-dimensionalizing-navier-stokes.html|publisher=CFD Online|accessdate=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
{{Refend}}
*Y. Cengel and J. Cimbala, FLUID MECHANICS: Fundamentals and Applications, 4th Edition, McGraw-Hill Education, 2018 (see p521, section 10.2. Nondimensionalized Equations of Motion).

==Further reading==
{{Refbegin}}
*{{cite book
| title = Applied Analysis of the Navier–Stokes Equations
| volume = 12
| series = Cambridge Texts in Applied Mathematics
| first1 = C.R.
| last1 = Doering
| author1-link = Charles R. Doering
| first2 = J.D.
| last2 = Gibbon
| publisher = Cambridge University Press
| year = 1995
| isbn = 9780521445689
| url-access = registration
| url = https://archive.org/details/appliedanalysiso0000doer
}}
*{{cite book
| last=Tritton
| first=D.J. 
|author1-link=David Tritton 
| chapter = Chapter 7 – Dynamic similarity 
| title = Physical fluid dynamics
| year = 1988
| publisher = Clarendon Press
|location = Oxford [England]
| isbn = 0198544898
| edition = 2nd
}}
*{{Cite book
| publisher = SIAM
| isbn = 9780898715941
| last1 = Mattheij
| first1 = R.M.M.
| first2 = S.W. 
| last2 = Rienstra
| first3 = J.H.M. 
| last3 = ten Thije Boonkkamp
| title = Partial Differential Equations: Modeling, Analysis, Computation
| year = 2005
| chapter = §7.4 – Scaling and Reduction of the Navier–Stokes Equations
| pages = 148–155
}}
*{{Cite book
| publisher = Academic Press
| isbn = 9780123708854
| last = Graebel
| first = William
| title = Advanced Fluid Mechanics
| url = https://archive.org/details/advancedfluidmec00grae
| url-access = limited
| year = 2007
| chapter = §6.2 – The Boundary Layer Equations
| pages = [https://archive.org/details/advancedfluidmec00grae/page/n187 171]–174
}}
*{{Cite book
| publisher = Cambridge University Press
| isbn = 9780521849104
| last = Leal
| first = L. Gary
| author-link = Gary Leal
| title = Advanced Transport Phenomena: Fluid Mechanics and Convective Transport Processes
| year = 2007
}}<br>This book contains several examples of different non-dimensionalizations and scalings of the Navier–Stokes equations, see p.&nbsp;430.
*{{Cite book
| publisher = John Wiley & Sons
| isbn = 9780471772613
| last = Krantz
| first = William B.
| title = Scaling Analysis in Modeling Transport and Reaction Processes: A Systematic Approach to Model Building and the Art of Approximation
| year = 2007
}}
*{{Cite book
| title = Asymptotic Modelling of Fluid Flow Phenomena 
| series= Fluid Mechanics and Its Applications 
| volume=64 
| last=Zeytounian 
| first=Radyadour Kh. 
| year=2002 
| publisher=Kluwer Academic Publishers
| isbn=978-1-4020-0432-2 
}}
{{Refend}}

{{DEFAULTSORT:Non-dimensionalization and scaling of the Navier-Stokes equations}}
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[[Category:Dimensional analysis]]
[[Category:Equations of fluid dynamics]]