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TECHNICAL PAPERS

On the Use of the Squire-Long Equation to Estimate Radial Velocities in Swirling Flows

[+] Author and Article Information
Michel J. Cervantes

Division of Fluid Mechanics,  Luleå University of Technology, SE-97187 Luleå, SwedenMichel.Cervantes@ltu.se

L. Håkan Gustavsson

Division of Fluid Mechanics,  Luleå University of Technology, SE-97187 Luleå, Sweden

J. Fluids Eng 129(2), 209-217 (Aug 07, 2006) (9 pages) doi:10.1115/1.2409331 History: Received May 31, 2005; Revised August 07, 2006

A method to estimate the radial velocity in swirling flows from experimental values of the axial and tangential velocities is presented. The study is motivated by the experimental difficulties to obtain this component in a draft tube model as evidenced in the Turbine-99 IAHR∕ERCOFTAC Workshop. The method uses a two-dimensional nonviscous description of the flow. Such a flow is described by the Squire-Long equation for the stream function, which depends on the boundary conditions. Experimental values of the axial velocities at the inlet and outlet of the domain are used to obtain the boundary conditions on the bounded domain. The method consists of obtaining the equation related to the domain with an iterative process. The radial velocity profile is then obtained. The method may be applied to flows with a swirl number up to about Sw=0.25. The critical value of the swirl number depends on the velocity profiles and the geometry of the domain. The applicability of the methodology is first performed on a swirling flow in a diffuser with a half angle of 3deg at various swirl numbers, where three-dimensional (3D) laser Doppler velocimeter (LDV) velocity measurements are available. The method is then applied to the Turbine-99 test case, which consists in a model draft tube flow where the radial inlet velocity was undetermined. The swirl number is equal to Sw=0.21. The stability and the convergence of the approach is investigated in this case. The results of the pressure recovery are then compared to the experiments for validation.

Copyright © 2007 by American Society of Mechanical Engineers
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Figures

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Figure 1

CAD drawing of the Höllefors draft tube (in millimeters) (1)

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Figure 2

Schematic of the computed area (11)

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Figure 3

Interpolated inlet axial velocity (line) and experimental values (symbol) at different swirl numbers (11). The square (r=0.05, Uinlet=0.71) represents the first assumption for the end of the profile on the wall for Sw=0.1.

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Figure 4

Interpolated inlet tangential velocity (line) and experimental values (symbol) at different swirl numbers (11)

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Figure 5

Interpolated outlet axial velocity (line) and experimental values (symbol) at different swirl numbers (11)

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Figure 6

Average value of the stream function over the domain function of the inverse triangle height (in percent). The average value of the stream function is normalized with the value obtained for the grid used to perform the calculation. The inverse triangle height is normalized with the minimum triangle height. Sw=0 and Sw=0.10 correspond to the test case of Sec. 3 (Dahlhaug (11)). Cases 1–3 correspond to the test case of section 4 (Turbine-99).

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Figure 7

Representation of the stream function for v,x=0 and Sw=0.1

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Figure 8

Outlet radial velocity obtained after three iterations for Sw=0 and Sw=0.10

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Figure 9

Residual for Sw=0 and Sw=0.10

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Figure 10

Representation of the stream function for v,x=0 and Sw=0.35

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Figure 11

Schematic of the computed area (1)

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Figure 12

Interpolated inlet axial velocity (line) and experimental values (symbol) (1). Case 1: third-order interpolation, cases 2 and 3: sixth-order interpolation.

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Figure 13

Interpolated inlet tangential velocity (line) and experimental values (symbol) (1). Case 1: third-order interpolation, cases 2 and 3: sixth-order interpolation.

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Figure 14

Interpolated outlet axial velocity (line) and experimental values (symbol) (1). Case 1: third-order interpolation, case 2: sixth-order interpolation, case 3: sixth-order interpolation with added values.

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Figure 15

Flow visualization below the cone (courtesy of Urban Andersson)

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Figure 16

Radial velocity profile proposed at the Turbine-99 workshop (3), obtained by (3) and calculated with the present method after seven iterations for cases 1–3

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Figure 17

Residual for test cases 1–3

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Figure 18

Pressure recovery along the outer-wall; experimental and obtained from cases 1–3 after seven iterations

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