Draft:Scale Analysis of Viscous Rotational Flow

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Scale analysis is a key method in fluid dynamics that simplifies the governing equations of motion by identifying the dominant physical effects in a given flow. For viscous rotational flows, the analysis focuses on the balance between viscosity, rotation, and inertia, leading to important dimensionless parameters like the Reynolds number, Rossby number, and Ekman number. These non-dimensional numbers reveal how different forces (such as inertial, viscous, Coriolis, and centrifugal forces) interact in rotating fluid systems, such as ocean currents, atmospheric circulations, and rotating machinery.

Governing equations for viscous rotational flow

The equations governing viscous rotational flow stem from the Navier-Stokes equations and the continuity equation, adapted to account for rotation. These equations describe the fluid's motion under the influence of forces such as pressure gradients, viscosity, and Coriolis effects.

Continuity equation (Conservation of Mass)

For an incompressible fluid, where density ρ is constant, the continuity equation is:

$\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}+\frac{\partial w}{\partial z}=0$

• Here, $u$, $v$, and $w$ are the velocity components in the $x$, $y$, and $z$ directions, respectively. This equation ensures mass conservation within the flow.

Navier-Stokes equation (conservation of momentum)

The Navier-Stokes equation for an incompressible fluid in a rotating reference frame, with angular velocity Ω, is given by:

$\rho (\frac{\partial 𝐮}{\partial t}+𝐮\cdot \mathrm{\nabla }𝐮)=-\mathrm{\nabla }p+\mu {\mathrm{\nabla }}^2𝐮-2\rho \mathrm{\Omega }\times 𝐮-\rho \mathrm{\Omega }\times (\mathrm{\Omega }\times 𝐫)$

• Where:
  • u =($u$,$v$,$w$) is the velocity vector,
  • ρ is the fluid density,
  • $p$ is the pressure,
  • μ is the dynamic viscosity,
  • $\mathrm{\Omega }$ is the angular velocity vector,
  • r is the position vector.

The terms represent the pressure gradient force $-\mathrm{\nabla }p$, viscous force $\mu {\mathrm{\nabla }}^2$$u$, Coriolis force $-2\rho \mathrm{\Omega }\times 𝐮$, and centrifugal force $-\rho \mathrm{\Omega }\times (\mathrm{\Omega }\times 𝐫)$.

Vorticity Equation

The vorticity equation, describing the evolution of the vorticity ω=∇×u, is particularly useful in rotational flows:

$\frac{\partial \mathit{\omega }}{\partial t}+𝐮\cdot \mathrm{\nabla }\mathit{\omega }=\mathit{\omega }\cdot \mathrm{\nabla }𝐮+\nu {\mathrm{\nabla }}^2\mathit{\omega }-2\mathrm{\Omega }\cdot \mathrm{\nabla }𝐮$

This equation highlights how vorticity is affected by vorticity stretching, viscous dissipation, and Coriolis forces.

Dimensionless Parameters in Scale Analysis

Non-dimensionalization of the governing equations reveals several dimensionless parameters, which are critical for understanding the balance of forces in viscous rotational flows.

Reynolds Number (Re)

The Reynolds number represents the ratio of inertial forces to viscous forces:

$Re=\frac{UL}{\nu }$

• Where:
  • $U$ is the characteristic velocity,
  • $L$ is the characteristic length,
  • $v=\frac{\mu }{\rho }$​ is the kinematic viscosity.

Implications:

• High $Re$ indicates that inertial forces dominate (leading to turbulence).
• Low $Re$ suggests viscous forces dominate (resulting in laminar flow).

Rossby Number (Ro)

The Rossby number measures the significance of rotational (Coriolis) forces compared to inertial forces:

$Ro=\frac{U}{2\mathrm{\Omega }L}$

• Where $\mathrm{\Omega }$ is the angular velocity of the rotating system.

Implications:

• Small $Ro$ values indicate that Coriolis forces are dominant, typical in large-scale geophysical flows.

Ekman Number (Ek)

Template:Main The Ekman number represents the ratio of viscous forces to Coriolis forces:

$Ek=\frac{\nu }{\mathrm{\Omega }{L}^2}$

Implications:

• A small $Ek$ indicates that rotational effects dominate over viscous forces, leading to the formation of thin boundary layers, known as Ekman layers.

Taylor Number (Ta)

The Taylor number applies to flows between rotating cylinders, comparing centrifugal forces to viscous forces:

$Ta=\frac{4{\mathrm{\Omega }}^2{L}^4}{{\nu }^2}$

Implications:

• High $Ta$ numbers indicate the onset of centrifugal instabilities and the formation of Taylor vortices.
• Low $Ta$ numbers imply stable laminar flow.

Froude Number (Fr)

The Froude number compares inertial forces to gravitational forces:

$Fr=\frac{U}{\sqrt{gL}}$

• Where g is the acceleration due to gravity.

Implications:

• The Froude number is especially important in free surface flows and other scenarios where gravity significantly affects the fluid motion.

Non-dimensional Navier-Stokes equation

By non-dimensionalizing the Navier-Stokes equation, the various physical forces can be expressed in terms of dimensionless numbers. The resulting non-dimensional Navier-Stokes equation is:

$\frac{\partial \widetilde{𝐮}}{\partial \tilde{t}}+\widetilde{𝐮}\cdot \mathrm{\nabla }\widetilde{𝐮}=-\mathrm{\nabla }\tilde{p}+\frac{1}{Re}{\mathrm{\nabla }}^2\widetilde{𝐮}-\frac{1}{Ro}\widehat{𝐳}\times \widetilde{𝐮}-\frac{1}{F{r}^2}\widehat{𝐠}$

• Where:
  • $\widetilde{𝐮}$,$\widetilde{𝐭}$ , and $\widetilde{𝐩}$​ are the non-dimensional velocity, time, and pressure,
  • $Re$, $Ro$, and $Fr$ correspond to the Reynolds, Rossby, and Froude numbers, respectively.

Example: Rotating Cylinder Flow

Consider the flow of fluid within a rotating cylinder of radius R, rotating with angular velocity Ω. In this scenario, the characteristic velocity is the tangential velocity U=ΩR, and the length scale is L=R. Performing scale analysis gives:

• Reynolds number:

$Re=\frac{\mathrm{\Omega }{R}^2}{\nu }$

• Rossby number:

$Ro=\frac{1}{2}$

• Ekman number:

$Ek=\frac{\nu }{\mathrm{\Omega }{R}^2}$

Interpretation:

• When $Re$ is large, turbulence may occur.
• Small $Ek$ indicates the formation of thin boundary layers influenced by rotation, known as Ekman layers.

Importance of Scale Analysis

Scale analysis allows for a deeper understanding of which forces dominate in a given flow, enabling the simplification of the governing equations. In many engineering and geophysical contexts, knowing the balance between viscosity, rotation, and inertia is crucial for modeling flows, predicting the formation of vortices, and identifying turbulence thresholds.

References

• Batchelor, G. K. (2000). An Introduction to Fluid Dynamics. Cambridge University Press. ISBN: 978-0521663961.
• Kundu, P. K., & Cohen, I. M. (2002). Fluid Mechanics. Academic Press. ISBN: 978-0123914272.
• Greenspan, H. P. (1968). The Theory of Rotating Fluids. Cambridge University Press. ISBN: 978-0521067213.
• Schlichting, H., & Gersten, K. (2017). Boundary-Layer Theory. Springer. ISBN: 978-3662537645.
• Tritton, D. J. (1988). Physical Fluid Dynamics. Clarendon Press. ISBN: 978-0198538959.