Draft tube

Application Challenge 6-07 © copyright ERCOFTAC 2004

Introduction

An experimental study has been performed of the flow in a sharp-heel draft tube for a Kaplan turbine. The measurements were made at the Turbine Testing Facility at Vattenfall Utveckling AB in Älvkarleby, Sweden. The draft tube is the curved diffuser downstream of the runner in a water turbine, and the purpose of the draft tube is to recover as much as possible of the kinetic energy of the flow after the runner. The efficiency loss in the draft tube is particularly important for low-head, high flow Kaplan power stations. The complete model consists of a 1:11 scale copy of the power station Hölleforsen including the rotating Kaplan runner, which has a diameter of 0.5 m. Figure 1.1 shows the draft tube. The Reynolds number ReD, based on runner diameter and average velocity is 1 750 000.

Figure 1.1. Schematic of Draft Tube Application Challenge.

Relevance to Industrial Sector

The flow in fluid machinery often involves the application of diffusers to convert kinetic energy in the fluid into pressure rise. A detailed understanding of the physics of the flow in such diffusers is of fundamental importance in order to improve the efficiency of the diffuser. The pressure recovery in a draft tube, which is the curved diffuser downstream of the runner in a water turbine, takes place in a very complicated geometry, which is determined by economic considerations regarding construction as well as fluid-mechanical considerations. Building costs are balanced against diffuser performance, and earlier designs (when building costs were relatively high) are optimised differently than modern designs. Whether the flow in the draft tube separates or not determines the ultimate pressure rise that can be achieved.

When refurbishing old hydro-power plants, there is therefore a potential for improving the pressure recovery in the draft tube by modifying the geometry of the draft tube. In order to utilise this potential, validated CFD tools are necessary.

The flow downstream of the runner in a water turbine, which is the inlet flow to the draft tube, is complicated with both a swirling mean flow, a periodic velocity component, and turbulence. The flow in the draft tube itself is composed by several co-existing flow phenomena or flow regimes: boundary layers in positive pressure gradient, flow curvature, flow rotation, and possibly (2- or 3-D) flow separation, thus creating a real application challenge. The test case put forward here is designed to test the ability of CFD codes to predict the pressure rise and flow field for such problems.

Design or Assessment Parameters

The main assessment parameters (DOAPs) for this AC are

• Pressure recovery factor Cpr

• Energy loss coefficient z

• Kinetic energy correction factors aaxial and aswirl

• Momentum correction factor b

• Swirl intensity S

The by far most important single assessment parameter is the pressure recovery factor.

Details of how to calculate each one of these are described below.

• Pressure recovery factor

The pressure recovery factor is defined as:

Cp_{r{\text{ wall}}}={\frac {P_{\text{out:wall}}-P_{\text{in:wall}}}{{\frac {1}{2}}\rho \left({\frac {Q}{A_{\text{in}}}}\right)^{2}}}\qquad \qquad \qquad \qquad (1)

where P_out:wall is the outlet averaged static wall pressure at cross section IVb (Fig. 1.3 and 1.4), P_in:wall is the inlet averaged static wall pressure at cross section Ia (Fig. 1.3 and 1.4), r is the density, Q is the flow rate (m³/s) and A_in is the area at cross section Ia (Fig. 1.3 and 1.4). Observe that A_in does not include the runner cone in the centre of the inlet.

The inlet averaged static wall pressure, P_in:wall, has bean found very sensitive to the inlet conditions and the geometry at the inlet. Therefore, an alternative Cp_r average

Cp_{r{\text{ average}}}={\frac {P_{\text{out:mean}}-P_{\text{in:mean}}}{{\frac {1}{2}}\rho \left({\frac {Q}{A_{\text{in}}}}\right)^{2}}}\qquad \qquad \qquad (2)

has also been used in the evaluation of the results, where P_out:mean and P_in:mean are the averaged static pressures across these sections. Note: P_out:mean for the experiments is estimated with P_out:wall, since the pressure has only been determined at the wall at the outlet section.

Cp_r average is a direct result of the whole field solution for all variables (if the pressure is set at the outlet) and much less sensitive to different changes at the inlet. Unfortunately it is more difficult to determine experimentally, resulting in larger uncertainty.

The pressure recovery factor indicates the degree of conversion of kinetic energy into static pressure where a higher value means higher efficiency for the draft tube. The exact value of the pressure recovery factor depends on the whole field solution and can be seen as an integral property of the solution.

• Energy loss coefficient

\zeta ={\frac {\iint \limits _{A1}p_{1}{\vec {u}}_{1}{\vec {dA}}_{1}-\iint \limits _{A2}p_{2}{\overline {u}}_{2}{\overline {dA}}_{2}+\iint \limits _{A1}\rho {\frac {|{\overline {u}}_{1}|^{2}}{2}}{\overline {u}}_{1}dA_{1}-\iint \limits _{A2}\rho {\frac {|{\overline {u}}_{2}|^{2}}{2}}{\overline {u}}_{2}dA_{2}}{\iint \limits _{A1}\rho {\frac {|{\overline {u}}_{1}|^{2}}{2}}{\overline {u}}_{1}dA_{1}}}\qquad \qquad \qquad (3)

where a lower value means higher efficiency for the draft tube. Index 1 refers to cross section Ia (Fig. 1.3 and 1.4) and index 2 refers to cross section IVb (Fig. 1.3 and 1.4). This coefficient is directly coupled to the losses in the system but is seldom used in experimental work since it requires knowledge about the variables over the whole inlet and outlet cross section. However, in CFD it is easily calculated.

• Kinetic energy correction factors aaxial and aswirl at cross section Ia and cross section III :

Failed to parse (syntax error): {\displaystyle \[\alpha_{axial} = \frac{1}{A\overline{u}^3_\alpha{x}} \int\limits_A u^3_\alpha{x} dA\]} (4)

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbases.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \[\alpha_{swirl} = \frac{1}{A\overline{u}^3_\alpha{x}} \int\limits_A u^2_{tang} u_\alpha{x} dA\]} (5)

True Kinetic energy flux = Failed to parse (syntax error): {\displaystyle \[{(\alpha_{axial} + \alpha_{swirl})}\]} . (Kinetic energy flux computed using mean velocity)

True specific kinetic energy: Failed to parse (syntax error): {\displaystyle \[e_{kin} = \frac{1}{2A\overline{u}_{\alpha x}} \left(\int\limits_A u^2 u_{\alpha x} dA\right) = \frac{1}{2A\overline{u}_{\alpha x}} \left(\int\limits_A u^3_{\alpha x} dA + \int\limits_A u^2_{tang} u_{\alpha x} dA\right)\]}

Specific kinetic energy is computed using mean velocity: Failed to parse (syntax error): {\displaystyle \[\overline{e}_{kin} = \frac{\overline{u}^2_{\alpha x}}{2}\]}

where A is the cross section area, u is the local magnitude of the total velocity vector, ${\overline {u}}_{\alpha x}$ is the mean axial velocity perpendicular to the cross section = Q/A, uax is the local axial velocity and utang is the local tangential velocity.

• Momentum correction factor b which is defined as :

Failed to parse (syntax error): {\displaystyle \[\beta = \frac{True momentum flux}{Momentum flux computed using mean velocity} = \frac{1}{A\overline{u}^2_{\alpha x}} \int\limits_A u^2 dA\]} (6)

where u is the local velocity, is the mean axial velocity and A is the cross sectional area.

All of aaxial, aswirl and b characterise the nonuniformity of the velocity profile.

• Swirl intensity S at cross section Ia and cross section III:

Failed to parse (syntax error): {\displaystyle \[S = \frac{\int\limits_A \left[u_{tang^r} {(pu_{\alpha x})}\right] dA}{p\int\limits_A u^2_{\alpha x} dA} = \frac{1}{R}\frac{\int\limits^{r0+R}_{r0} u_{\alpha x} u_{tang^{r^2}} dr}{\int\limits^{r0+R}_{r0} u^2_{\alpha x^{r}} dr} = \frac{angle momentum flux}{R \cdot axial momentum flux}\]} (7)

where uax is the axial velocity, utang the tangential velocity, r0 the inner radius and r0+R the outer radius (see Fig. 1.5).

Still another pressure coefficient can be defined as

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbases.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \[C_p' = \frac{\int\limits_{A2} pu_{\alpha x} dA - \int\limits_{A4} pu_{\alpha x} dA}{\frac{1}{2}\rho\left(\frac{Q}{A_1}\right)^2 Q}\]} (8)

in terms of which the loss coefficient, z, can be written as

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbases.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \[\zeta = 1- \frac{\alpha_2}{\alpha_1} \left(\frac{A_1}{A_2}\right)^2 - \frac{C'_p}{\alpha_1}\]} (9)

where a1 and a2 are the sum of the swirl components at inlet and outlet, respectively. Since the last term in this expression is close to unity, the evaluation of z is ill-conditioned, a fact that will be discussed in some detail when assessing the accuracy of z.

Flow Domain Geometry

The flow geometry associated with the AC is specified below. The geometry is also available in different CAD-formats at [http://www.sirius.luth.se/strl/Turbine-99/index.htm]. At the project home page some additional geometric information and a link to a ‘start’ grid that can be used for a grid-independent check of the CFD-numerics.

Figure 1.2 is a drawing of the entire draft tube. In Figure 1.3 one can see a sketch of available measurements at different cross sections, and in Figure 1.4 and Figure 1.5 exact locations for measurements for each cross section are given. Figure 1.6 defines the co-ordinate system used. Figure 1.7 shows the location of the upper and lower centre lines of the draft tube.