Vortex Elements

Vortex Theory Manual

The vortex element applies physical equations that relate static pressure, total temperature, and fluid velocity at the vortex inlet with those at the vortex exit. It basically does the Pressure ratio and Temperature rise calculations based on the type of Vortex Element (Free Vortex, Forced Vortex, Forced Vortex with Cavity, Forced Vortex on Rotor, Cavity Vortex, Generic Vortex). For both the free vortex and the forced vortex, the assumption is made that the flow through the vortex is isentropic and that fluid properties are adequately represented by the specific heat ratio and the fluid gas constant as determined at the vortex inlet temperature & pressure. In the equations presented in this section, the subscript 1 denotes the vortex segment inlet, and the subscript 2 denotes the vortex segment exit. The inlet and exit of each vortex segment are referred to as “stations” in output. For a multi-segment vortex, the vortex parameters (Swirl, Pressure rise & Temperature gradients) are computed for each vortex segment and assembled for the entire segmented vortex. If there is flow reversal, then the inlet and exit radii trade places; the inlet swirl factor, pressure, and temperature are generally changed as well. However, the same vortex equations are applied.

Nomenclature:

Subscripts:

Vortex Exit Velocity Calculations

The following table provides details about how the axial, radial, and tangential components of the fluid velocity at the vortex exit are calculated. The equation(s) used to calculate tangential component of the fluid velocity at the vortex exit are determined by the type of vortex. The tangential exit velocity directly effects the vortex pressure ratio and vortex temperature rise calculations.

VELOCITY COMPONENT	VORTEX TYPE (P_RATIO_FLG / DELTA_T_FLG)	EQUATION(S)
Axial	ALL
Radial	ALL
Tangential	FREE VORTEX
	FORCED VORTEX
	CAVITY VORTEX	= velocity of air exiting cavity (See cavity section for information)

Note: Vortex Radial and Axial Velocity Attenuation

Full Attenuation: (default): None of the axial and radial velocity passes through the vortex element to the downstream chamber. The axial and radial velocity is not driving flow through a downstream restriction.

No Attenuation: All of the axial and radial velocity passes through the vortex element to the downstream chamber. The axial and radial velocity is available to drive flow through a downstream restriction.

Vortex Pressure Ratio & Temperature Rise Calculations

Free Vortex

The Pressure ratio for Free Vortex are derived from radial equilibrium equation & the final form is given as follows

For Compressible Gas:

For Incompressible Gas & Liquids:

There is no change in total temperature in an isentropic free vortex. In this case, the change in static temperature is opposite and equal to the change in the dynamic temperature resulting from the change in the tangential velocity. Thus, for an isentropic free vortex we obtain

For more details about the derivations of Pressure ratio & temperature rise refer Section 2.3 of “Gas Turbines Internal Flow Systems Modeling” by Bijay Sultanian

Forced Vortex

The Pressure ratio & Temperature Rise for Forced Vortex are given as follows

For Compressible Gas:

For Incompressible Gas & Liquids:

The change in total temperature in an isentropic forced vortex equals twice the change in its static temperature

For more details about the derivations of Pressure ratio & temperature rise refer Section 2.3 of “Gas Turbines Internal Flow Systems Modeling” by Bijay Sultanian

Forced Vortex on the Rotor

This vortex subtype is same as Forced vortex but differ only in calculating Temperature rise across vortex. For Forced Vortex on Rotor, change in temperature will be based on XK = 1, independent of XK specified for the vortex

.

Cavity Vortex

For the cavity vortex, the solution of fluid swirl velocity and vortex temperature rise is done in the cavity solver. Calculation of vortex pressure ratio is done in the cavity solver routine if the cavity is associated with a single vortex with all inflow at one radius and all outflow at another radius. If several different vortexes with various inflow and outflow radii are involved in a cavity, the cavity solver determines the swirl velocity of the fluid in the cavity and a single temperature rise from the mixed inlet temperature to an average fluid exit total absolute temperature. The vortex pressure ratio for each vortex is then calculated external to the cavity solver assuming an isothermal vortex at the cavity average static temperature as given below

For Compressible Gas:

For Incompressible Gas & Liquids:

The change in total temperature in an isothermal vortex is given by

More information about cavity XK calculations & cavity windage are available in Cavity Theory Manual.

Forced Vortex (Cavity if Present)

The pressure ratio & temperature rise across vortex is calculated based on equation specified in Cavity Vortex subtype if cavity is associated with chambers else Forced vortex subtype is used

Generic Vortex

Apart from the options which are presented above (Free, Forced, Cavity), the generic vortex provides two more options for users to calculate Vortex pressure rise across any vortex segments. They are

1. P ratio equals 1

   

2. User Defined: User specified Pressure ratio for vortex segment
   For Forward Flow:

   

   For Reverse Flow:

   

For the temperature rise calculations, in addition to Free, Forced, Cavity options, the generic vortex provides 3 more options as mentioned below

1. Delta T = 0
   For Inertial or Non-Rotating Frame:
   For Rotating Frame:
2. User Defined:
   For Inertial or Non-Rotating Frame:
   For Forward Flow:

   

   For Reverse Flow:

   

   For Rotating Frame
   For Forward Flow:

   

   For Reverse Flow:

   

3. DT_PUMPING

   

Note: Temperature Rise calculation Options

In Some instance, Flowsimulator automatically modifies Vortex Temperature rise calculations functions to prevent errors related to mismatch between user specified options and the swirl chamber.

For Example:

1. If the user specifies Vortex Temperature rise calculations as cavity ΔT (DELTA_T_FLG = 3) and the supplied swirl chamber is not an inertial-type or vortex-type chamber or the swirl chamber does not have an associated cavity, FlowSimulator will use the DT_PUMPING for calculating temperature rise across vortex segments (DELTA_T_FLG = 8)
2. If the user specifies Vortex Temperature rise calculations as cavity ΔT (DELTA_T_FLG = 3) and the supplied swirl chamber is inertial with an associated cavity, FlowSimulator will change the Temperature rise calculations to Fixed Delta T = 0 and the cavity windage is applied directly to the inertial chamber.

Temperature frame conversions

As discussed above, both the vortex inlet temperature and the vortex temperature rise depend on the Frame of reference, the chamber type to which the vortex is connected, and the rotational speed of the element with which the vortex is associated. As discussed in Chambers section, the chamber is assumed to be rotating at the speed of the element if its type is either Plenum or Momentum, while it is stationary if the chamber type is Inertial. If chambers at either end of the vortex store temperature in a different temperature frame, then temperature frame conversions are required. The following table shows the equations used when temperature frame conversions are required at the vortex inlet, exit, or both.

CONVERSION TYPE	LOCATION	EQUATION
REL_TO_ABS	INLET
	EXIT
ABS_TO_REL	INLET
	EXIT

Vortex Flow Rate

Unlike other elements in FlowSimulator, there is no governing equation for determining Mass flow rate across Vortex. The flow rate in a vortex is determined by the flow imbalance in the chambers at each end of the vortex. The flow imbalance does not include the vortex element flow. Flow (W) entering the chamber is positive and flow exiting the chamber is negative

By Default, the Automatic Option is used to determine the flow, where solver searches for the best vortex end to use for the imbalance. There are some user controls over the method used as described below

1. Automatic – (default) solver searches for the best vortex end to use for the imbalance.
2. Average Up and Down – always uses both ends of vortex

   

3. Upstream – always uses upstream chamber of the vortex
4. Downstream – always uses the downstream chamber of the vortex
5. Other – Uses an element (User should type in number of a more stable element number in series with the vortex element)