# Laminar Free Surface Flow Inside a Rotating Cavity

In this application, AcuSolve is used to solve for the flow field within a rotating cavity with the top of the cavity free to move as if it was exposed to air. The height of the free surface is determined and compared against the analytical solution for the same rotational velocity under the standard gravitational force.

## Problem Description

The problem consists of a fluid with arbitrary material properties rotating within an axisymmetric cavity. The cavity is 1.0 m in height with a radius of 0.5 m and is rotating at a constant speed of 3.0 rad/sec, as shown in the following image, which is not drawn to scale. The model is simulated as transient with the fluid initially at rest within the cavity. The radial velocity applied to the cavity walls causes the fluid's free surface to creep up the side wall forming a parabolic shape. The height of this surface is compared against the analytical solution as described in White, 2003.

The problem is simulated as axisymmetric by considering a 30 degree portion of the cavity with rotationally periodic boundary conditions applied on the planar surfaces representing the cut faces of the cavity.

## AcuSolve Results

The AcuSolve solution reaches a quasi-static result with the free surface height stabilizing, demonstrating the final flow conditions of the simulation. As the fluid within the cavity is subjected to the multiple reference frame simulating the rotating boundaries the free surface begins to deform. At the center of the cavity the fluid height decreases and in order to conserve mass the height steadily increases as a function of the radial distance from the center. The surface height takes the form of a parabola as described in White, 2003. The following image demonstrates the shape that the fluid forms within the cavity, showing the vertical displacement of the rotating fluid. The free surface height is compared with the analytical solution where the image shows black circles representing the analytical solution and a solid red line for the AcuSolve results.

## Summary

The AcuSolve solution compares well with analytical results for laminar free surface flow inside a rotating cavity. In this application, a fluid initially at rest is subjected to an angular velocity about the center of the cavity. As a result of the prescribed velocity, the fluid is forced to the outer wall of the cavity and develops a free surface profile in the shape of a parabola. The AcuSolve solution for the free surface height matches nearly exactly with the analytical results, with only minor differences based on the convergence criteria that you specified.

## Simulation Settings for Laminar Free Surface Flow Inside a Rotating Cavity Fluid

Global

• Problem Description
• Analysis type - Transient
• Flow equation - Navier Stokes
• Turbulence equation - Laminar
• Mesh type - Arbitrary Mesh Movement (ALE)
• Auto Solution Strategy
• Final time - 15.0 sec
• Initial time increment- 0.05 sec
• Convergence tolerance - 0.001
• Min stagger iterations - 0
• Max stagger iterations - 4
• Material Model
• Fluid
• Density
• Type - Constant
• Density - 10 kg/m3
• Viscosity
• Type - Constant
• Viscosity - 3.0 kg/m-sec
• Reference Frame
• Omega
• Rotation Center
• X-coordinate - 0.0 m
• Y-coordinate - 0.0 m
• Z-coordinate - 0.0 m
• Angular velocity
• X-component - 0.0 rad/sec
• Y-component- 0.0 rad/sec
• Z-component - 3.0 rad/sec

Model

• Volumes
• Fluid
• Element Set
• Material model - Fluid
• Body force - Gravity
• Reference frame - Omega
• Surfaces
• + Y
• Simple Boundary Condition - (disabled to allow for nodal boundary conditions)
• Nodal Boundary Conditions
• X-velocity
• Type - Constant
• Reference frame - Omega
• Constant value - 0.0
• Y-velocity
• Type - Constant
• Reference frame - Omega
• Constant value - 0.0
• Mesh X-Displacement
• Type- Zero
• Mesh Y-Displacement
• Type- Zero
• -Z
• Simple Boundary Condition
• Type - Wall
• Reference frame - Omega
• Mesh displacement BC type - Fixed
• Axisymmetric +X
• Simple Boundary Condition (disabled to allow for nodal boundary conditions)
• Axisymmetric -X
• Simple Boundary Condition (disabled to allow for nodal boundary conditions)
• Free_Surface
• Simple Boundary Condition
• ⃣ Type - Free Surface
• Periodics
• Axisymmetric Condition
• Periodic Boundary Condition
• Type - Axisymmetric
• Rotation Axis
• Point 1
• x-coordinate - 0.0 m
• y-coordinate - 0.0m
• z-coordinate - 0.0m
• Point 2
• x-coordinate - 0.0 m
• y-coordinate - 0.0m
• z-coordinate - 1.0m
• Nodes
• All
• Mesh X-Displacement
• Type - Zero
• Mesh Y-Displacement
• Type - Zero
• Centerline
• X-Velocity
• Type - Zero
• Y-velocity
• Type - Zero
• Mesh X-Displacement
• Type - Zero
• Mesh Y-Displacement
• Type - Zero

## References

White, F. Fluid Mechanics. New York: McGrawHill. 2003