# Rouse Number

on . Posted in Dimensionless Numbers

Rouse number, abbreviated as P or Z, a dimensionless number, is used in fluid dynamics and engineering to characterize the relative importance of buoyancy forces to viscous forces in a fluid flow.  The Rouse number is typically used in the context of fluid flows that involve particles or substances suspended in the fluid, such as sediment particles in water or solid particles in a gas.  It helps determine whether buoyancy or viscous forces dominate the behavior of these suspended particles.

### Rouse number Interpretation

• Ro < 1  -  In this regime, viscous forces dominate, and particles tend to follow the fluid streamlines closely.  This is often referred to as the "settling" regime.
• Ro ≈ 1  -  When the Rouse number is close to 1, the buoyancy and viscous forces are roughly balanced, and particles exhibit mixed behavior between settling and being carried by the fluid.
• Ro > 1  -  In this regime, buoyancy forces dominate, and particles tend to rise in the fluid against the direction of gravity.  This is sometimes referred to as the "upward transport" regime.

The Rouse number is particularly important in environmental engineering and sediment transport studies, where it helps predict the behavior of particles in rivers, lakes, oceans, and other natural bodies of water.  It's a valuable tool for understanding how sediment particles are transported, settle, or remain suspended in a fluid under various conditions.

### Rouse number formula

$$P \;=\; w_s \;/\; \beta \; k \; u*$$     (Rouse Number)

$$w_s \;=\; P \; \beta \; k \; u*$$

$$\beta \;=\; w_s \;/\; P \; k \; u*$$

$$k \;=\; w_s \;/\; P \; \beta \; u*$$

$$u* \;=\; w_s \;/\; P \; \beta \; k$$

Symbol English Metric
$$P$$ = Rouse number $$dimensionless$$
$$w_s$$ = sediment fall velocity $$ft \;/\; sec$$ $$m \;/\; s$$
$$\beta$$  (Greek symbol beta) = constant that correlates eddy viscosity to eddy diffusivity  (1) $$dimensionless$$
$$k$$ = von Karman constant  (0.40) $$dimensionless$$
$$u*$$ = shear velocity $$sec \;/\; ft^{\frac{1}{3}}$$ $$s \;/\; m^{\frac{1}{3}}$$

Tags: Flow Fluid Buoyancy