Cavities

Cavity General Description

The rotating cavity solver is used to calculate the fluid rotational velocity and temperature change in control volumes bounded by rotating (and stationary) surfaces. Cavities are commonly needed to model the secondary flows (flow not in the main gas path) of turbo machinery applications. The solver uses an angular momentum balance between the flow entering and leaving the control volume and the torque applied to the flow by the rotating and stationary surfaces. The rotating fluid leads to a pressure gradient in the radial direction. The rotating velocity (swirl) of the fluid is used by the vortex elements to introduce the pressure gradient into the flow model. The temperature change due to the cavity is automatically applied to the flow.

Quick Guide for Creating Cavity in GUI

Right click on a vortex chamber or inertial chamber (a cavity cannot be applied to a plenum or momentum chamber) and select “Create Cavity”. An optional cavity title can be given to help identify it. Next create the surfaces of the cavity. Cavities usually have multiple surfaces with each surface associated with a different part in the machine. Each surface must have a unique speed so do not use points from a rotating and stationary part on the same cavity surface.

Figure 1.

The “Append” option works best for creating a new surface. The other options can be used to edit an existing surface. Select the surface points on the screen to fully define the cavity surface. The geometry from an IGES file is useful for locating the surface points. The friction options for the surface can be modified if needed. Bolt Rows or other features that affect the surface drag can be included.

Figure 2.

After the surfaces are created, the flows entering and exiting the control volume must be defined. The “Auto Fill” button can be used to populate the table. However, “Auto Fill” must have information from completed elements to work properly, so complete input for elements that are attached to the cavity chamber before attempting “Auto Fill”. Manual edits to this table may be required. The 3 inputs for each flow:

1. Radius: The radius (inches) of the flow entering or exiting the cavity control volume.
2. Swirl Source: The chamber containing the swirl of the flow entering the cavity. Enter a “0” if the swirl of the fluid exiting the Flow Element is to be used (needed if no chamber contains the proper swirl). The Swirl Source for Outflows will only be used by the solver if the flow reverses in the element.
3. Flow Element: The number of the flow entering or exiting the cavity control volume.

Figure 3.

The Cavity section in the “Flow Model Browser” can be used to adjust the cavity display and to delete cavities if needed. The “Colour by surface rotor speed” is useful for visualizing the surface speeds (Blue = stationary and Red = Rotor 1). Left click twice on a cavity in the list to center the view on the cavity. The “User Settings” panel has the surface point size control as a percentage of the chamber/element symbol size.

Figure 4.

Cavity Inputs

The Flow Simulator cavity solver input data consists of four separate name list blocks near the bottom of the model database file. The first of these, $CAV1, defines the number of separate cavities to be analysed and is used to set any global input variables that are related to cavities. Each cavity requires a $CAV name list block and a $FLOWCAV name list block. Both $CAV and $FLOWCAV define cavity variables with the $FLOWCAV variables providing information about flow, swirl, pressure, and temperature connectivity between the cavity and chambers in the model. Each $CAV name list block contains NSURF, which identifies the number of surfaces in the cavity or sub-cavity. NSURF is the number of sets of $SURF name list blocks that are required for the cavity or sub-cavity since the $SURF name list block provides details about cavities surfaces. The following listing contains a description of the cavity inputs for each of the name list blocks. All required name list variables have been highlighted.

Cavity Theory Manual

The fluid rotational speed in the cavity is computed based on inlet velocities, mass flows, and computed surface torque. The surface torque and cavity swirl factor are interdependent. The cavity solver carries out an iterative procedure to ensure both sets of equations are satisfied. In particular, the Regula Falsi method is employed with the initial guess for cavity swirl to iteratively seek the correct cavity swirl value that satisfies the angular momentum equation within the specified tolerance. For the first call to the cavity solver, the initial guess is based on the user’s entry from upstream chamber swirl (or downstream chamber swirl if flows are reversed). For subsequent cavity runs, the initial guess is based on the previous cavity analysis result. The calculated cavity swirl is representative of the cavity fluid control volume and is used as the cavity exit swirl.

In Flow Simulator, vortices and cavities follow the chamber-centric modelling approach. Cavities can be modelled with multiple inflows and outflows. This section presents the physical equations that govern the cavity vortex as implemented in Flow Simulator. The feature that distinguishes the cavity vortex from free and forced vortexes is its calculation of torque caused by the rotating metal parts acting on the fluid passing through the cavity. Torque affects fluid velocity, temperature, and pressure in the cavity, which are subsequently calculated by including the torque term in the momentum and energy balance equations.

The Angular Momentum equation

Nomenclature:

KE: Volumetric Kinetic Energy

ρ: Fluid density

r, R: Radius from the engine centreline

Ω: Rotational speed

N: Speed in terms Rotation per Minute

XK: Ratio of the fluid absolute rotational speed to the rotational reference frame (i.e. rotor)

τ: Torque

cf: Coefficient of friction

x: Axial location of surface segment endpoint

α: Angle that the surface makes with the engine centreline

Nb: Number of bolts in the bolt row

Cdb: Cd for the bolt row (spacing correction is automatic)

Sb: Bolt interference scale factor for the bolt row

Ab: Cross sectional area of exposed bolt head surface

Rb: Radius from the engine centerline the bolt row

k: Cavity surface index

csf: Cylindrical surface skin friction

nsf: Non-cylindrical surface skin friction

CAV: Cavity control volume

ref: Cavity rotational reference frame

s: Cavity surface segment index

1,2: Cavity surface segment endpoints

i, j: Cavity inflow/outflow element indexes

const: Indicates that any radius terms have been removed from the parameter, this is useful to allow parameters to be pulled outside of an integral taken with respect to radius as “constants”

Surface Skin Friction Torque:

Torque due to non-cylindrical surfaces (Conical, Vertical surfaces with respect to engine centreline) and Cylindrical surfaces (Horizontal surfaces with respect to engine centreline) is computed either by a numerical integral or by the exact integral method. The torque integration method option is available under SolveràAnalysis SetupàSolver ConvergenceàCavity Solver Tolerancesà Torque integration method dropdown menu. The torque coefficients equations are summarized under following table:

The Volumetric kinetic energy term is defined as

Surface Bolts Form Drag Torque

Torque equation for bolts can be found in Sultanian (ref 6), section 4.3.2.

The interference scale factor, S_b,k,r is based on Fig 30 on page 5-14 of Hoerner (ref 7).

The skin friction coefficient equation depends on whether the surface is stationary or rotating. The following table presents the skin friction coefficient equations

The relation for friction coefficients on non-cylindrical surfaces that is used in all the cavity solvers was developed from data for rotating disks. Since most of the non-cylindrical surfaces in the cavities of interest are annuli rather than complete disks, use of an average radius was selected as the preferable method for the Flow Simulator cavity solver. As verified by running a wide range of test cases, the variation in results due to these methodological differences are generally minor and well within the range of engineering accuracy for the analysis.

Delta Temperature across cavity:

The temperature rise across the cavity is calculated using energy balance accounting for the windage generated by rotor. The Energy balance and Delta Temperature Rise calculation are as follows:

Single In/Outflow Cavity:

Multiple In/Outflow Cavity

Cavity Heat Transfer

HTC = Heat transfer coefficient (user-supplied) or calculated by the correlation.

Area = Surface area

T_surf = The surface temperature (user-supplied).

T_{fluid rel} = The relative fluid total temperature, adjusted for fluid and surface rotation.

Cavity Pressure

The cavity pressure is simply the pressure that Flow Simulator has assigned to the cavity chamber. If the cavity chamber is an inertial chamber the pressure is calculated during the normal network solution process. If the cavity chamber is a vortex chamber, the pressure comes from the vortex element segment that uses this vortex chamber. The pressure is only used in the cavity solver to determine the fluid properties. The pressure ratio across a cavity can only be calculated using a vortex element.

Cavity Surface Axial Force

Force_dir=Force direction (user-supplied) for this surface, either -1 or 1.

P_chamber=Static pressure of the cavity chamber.

Area = Surface area for this cavity surface, $Area=\pi *abs\left(radiu{s}_1^2-radiu{s}_n^2\right)$ .

Radius₁ = Radius of first point of cavity surface.

Radius_n = Radius of the last point of the cavity surface.

The cavity forces are summed for all cavities using the surface rotation and written to the *.cav.out file. For example, all forces are summed for the surface rotating with rotor 1 RPM.