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Scale Analysis of Navier-Stokes Equation

Introduction

Scaling analysis is a powerful analytical tool used to estimate the relative magnitudes of different physical effects in mathematical models, particularly in fields like aerodynamics, thermodynamics, and nonlinear dynamics. It involves simplifying complex equations. Instead of finding exact solutions, scaling analysis gives approximate order-of-magnitude predictions that serve as benchmarks for more detailed numerical or computational studies.

Navier-Stokes Equation

The Navier-Stokes equations for a two-dimensional incompressible flow in Cartesian coordinates (x,y) can be written as a system of equations that describe the conservation of mass (continuity equation) and the conservation of momentum in both the x- (streamwise) and y-directions.

The continuity equation for incompressible flow (where the density ρ is constant) is:

${\frac {\partial u}{\partial x}}+{\frac {\partial v}{\partial y}}=0$ Eq.(1)

where:

• u is the velocity component in the x-direction (streamwise),
• v is the velocity component in the y-direction (cross-stream).

The streamwise scalar component of the momentum equation in the x-direction is derived from the vector momentum equation:

${\displaystyle \rho \left({\frac {\partial u}{\partial t}}+u{\frac {\partial u}{\partial x}}+v{\frac {\partial u}{\partial y}}\right)=-{\frac {\partial p}{\partial x}}+\mu \left({\frac {\partial ^{2}u}{\partial x^{2}}}+{\frac {\partial ^{2}u}{\partial y^{2}}}\right)}$ Eq.(2)

$\underbrace {} _{A}$ $\underbrace {} _{B}$ $\underbrace {} _{C}$ $\underbrace {} _{D}$ $\underbrace {} _{E}$ $\underbrace {} _{F}$

where:

• ρ is the fluid density,
• p is the pressure,
• μ is the dynamic viscosity.

Assumed Orders of Magnitude for Derivatives

To begin scaling analysis, the spatial and time derivatives in the Navier-Stokes equations are assumed to have characteristic scales:

1. Streamwise Derivative:

${\frac {\partial }{\partial x}}\sim {\frac {1}{D}}$ Here, D is a characteristic length in the streamwise (x-) direction, such as the chord length of an airfoil or wing. The idea is that changes in the streamwise direction occur over a distance comparable to D.

1. Time Derivative:

$\quad {\frac {\partial }{\partial t}}\sim \omega$ Here, ω represents a characteristic frequency of unsteady oscillations in the flow. This frequency may either be prescribed based on known physical phenomena (such as vortex shedding or oscillations of a wing) or determined as an outcome of the scaling analysis.

1. Transverse Derivative:

$\quad {\frac {\partial }{\partial y}}\sim {\frac {1}{\lambda }}$ Here, λ is a characteristic length in the transverse (y-) direction. Unlike D, λ is not known at the outset but is often determined through scaling analysis itself. In boundary layer problems, for example, λ could represent the thickness of the boundary layer, which is typically much smaller than D.

Balancing Terms (A) and (B):-

So, for example, if one requires that terms (a) and (b) balance in Eq. 2, then

${\hat {u}}{\hat {w}}\sim U_{\infty }{\frac {\hat {u}}{D}}\rightarrow {\frac {\omega D}{U_{\infty }}}\sim 1$ Eq.(3)

Equation 3 aligns with a wide range of experimental evidence, including the classical oscillating flow in the wake behind blunt bodies (e.g., Von Kármán vortex street) and more recent findings in transonic buffeting flows. Accepting this equation indicates that terms (a) and (b) are balanced and equally significant for such flows. Additionally, Equation 3 allows for the determination of one parameter in relation to another, a well-established concept that has been beneficial for engineers and scientists for many years.

Balancing Terms (C) and (B):-

From this balancing it is determined that

$U_{\infty }{\frac {\hat {u}}{D}}\sim {\frac {{\hat {v}}{\hat {u}}}{\lambda }}\quad \rightarrow \quad {\frac {\hat {v}}{U_{\infty }}}\sim {\frac {\lambda }{D}}$ Eq.(4)

But recall is not yet known and is yet to be determined.

Balancing Terms (E) and (B):-

$\rho _{\infty }U_{\infty }{\frac {\hat {u}}{D}}\sim \mu {\frac {\hat {u}}{D^{2}}}\quad \rightarrow \quad Re\equiv {\frac {\rho _{\infty }U_{\infty }D}{\mu }}\sim 1$ Eq.(5)

The interpretation of this result is that when the Reynolds number (Re) is around one, terms (b) and (e) are balanced and equally significant. However, if Re is much greater than one, term (b) dominates, allowing term (e) to be neglected. This latter scenario is particularly relevant in aerospace applications, warranting further consideration.

Balancing Terms (F) and (B):-

From this balancing one determines that

$\rho _{\infty }U_{\infty }{\frac {\hat {u}}{D}}\sim \mu {\frac {\hat {u}}{\lambda ^{2}}}$ Eq.(6)

and solving Eq. (6) one determines that

${\frac {\lambda }{D}}\sim {\frac {1}{\sqrt {Re}}}$ Eq.(7)

Note that Equation 7 provides the desired estimate for . For Re≈1 and D≈1, Equation 7 determines for Re≈1. It's important to emphasize that Equations 6 and 7 are order of magnitude estimates rather than conventional equations. If Re is much greater than one, then is much less than D.

Balancing Terms (D) and (B):-

From this one determines that

$\rho _{\infty }U_{\infty }{\frac {\hat {u}}{D}}\sim {\frac {p}{D}}$ Eq.(8)

But how to estimate u? From $u=U_{\infty }+{\hat {u}}\quad {\text{and}}\quad \rho =\rho _{\infty }$ and considering term (b), it is clear that when the ow oscillates in a nonlinear limit cycle oscillation (in the absence of any structural body motion)

${\hat {u}}\sim U_{\infty }$ Eq.(9)

when the two components of term (b) are comparable, i.e.

$\left(U_{\infty }+{\hat {u}}\right){\frac {\partial {\hat {u}}}{\partial x}}=U_{\infty }{\frac {\partial {\hat {u}}}{\partial x}}+{\hat {u}}{\frac {\partial {\hat {u}}}{\partial x}}$ Eq.(10)

are comparable. Using Eq. 10 in Eq. 9, then

${\hat {p}}\sim \rho _{\infty }U_{\infty }^{2}$ Eq.(11)

and thus

$C_{L}\approx {\frac {2{\hat {p}}D}{\rho _{\infty }U_{\infty }^{2}D}}\approx 1$ Eq.(12)

Note that Equation 12 is a prediction, and factors of 2 are ignored in the order of magnitude (scaling) analysis. A computational solution to the Navier-Stokes equations is expected to yield a lift coefficient of order one for the oscillating lift on a blunt body.

References

1) White, F. M. (2016). Fluid Mechanics (7th ed.). McGraw-Hill Education.

2) Anderson, J. D. (2010). Fundamentals of Aerodynamics (5th ed.). McGraw-Hill Education.

3) Kundu, P. K., & Cohen, I. M. (2004). Fluid Mechanics (4th ed.). Academic Press.

Article Prepared By:

1) Aryan Bhargava (Roll No. 21135028), IIT BHU (Varanasi)

2) Aryan Kumar (Roll No. 21135029), IIT BHU (Varanasi)

3) Tanmay Shukla (Roll No. 21135136), IIT BHU (Varanasi)

4) Swayam Prakash (Roll No. 21135135), IIT BHU (Varanasi)

5) Sumit Ghosh (Roll No. 21135133), IIT BHU (Varanasi)