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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. 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.

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:

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

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

Substituting the scales the non-dimensionalized equation obtained is:

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. 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

Scaling of equation (Template:EquationNote) 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.Potter or Ceramic Artist Truman Bedell from Rexton, has interests which include ceramics, best property developers in singapore developers in singapore and scrabble. Was especially enthused after visiting Alejandro de Humboldt National Park. At low pressures the same equation reduces to a diffusion equation

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

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

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

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:

${\frac {\partial W}{\partial t}}+U{\frac {\partial W}{\partial X}}+W{\frac {\partial W}{\partial Z}}\ =-{\frac {1}{\rho _{o}}}{\frac {\partial p_{d}}{\partial Z}}+v\left({\frac {\partial ^{2}W}{\partial X^{2}}}+{\frac {\partial ^{2}W}{\partial Z^{2}}}\right)\ -g\left(\beta _{s}\nabla {S}-\beta _{T}\nabla {T}\right)$

Where S is the salinity of the fluid, β_T is the thermal expansion coefficient at constant pressure and β_S 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 S_T, T_T denote the salinity and temperature at top layer, S_B, T_B denote the salinity and temperature at bottom layer, Ra is the Rayleigh Number, and Pr is the Prandtl Number. The sign of Ra_S and Ra_T will change depending on whether it stabilizes or destabilizes the system.

References

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• Template:Cite web
• 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

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• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
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• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
  This book contains several examples of different non-dimensionalizations and scalings of the Navier–Stokes equations
• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534