Experimental analysis and modeling of the losses in the tip leakage flow of an isolated, non-rotating blade setup

Abstract

Improving pressure rise capabilities of axial compressors requires an in-depth understanding of the losses produced in the tip leakage region. Here, a generic setup that magnifies the tip region of an isolated, non-rotating blade is used with the objectives of describing the main flow components and evaluating the related sources of loss. The flow at the tip is structured by the jet flow out of the gap which, under the effect of the main stream, rolls-up into a tip-leakage vortex. The current setup is characterized by the tip gap height and the thickness of the incoming boundary layer at the casing, here a flat plate, for a given incidence of the blade. Measurements are performed using LDV and a multi-port pressure probe. Variations in the tip-leakage flow are found to be mainly driven by gap height. A small, intermediate and large gap regimes are more specifically found, with threshold around 4% and 8% of gap to chord ratio for the present setting. The incoming boundary layer thickness is shown to provoke a notable effect on the vortex lateral position and total pressure losses. The local entropy creation rate is computed from LDV data and used to identify the sources of loss in the flow. A decomposition into wake and vortex losses is further proposed, allowing to relate the contributions of the various flow components to the overall losses. An empirical model of the formation of the tip vortex is developed to account for the increased losses as a function of gap height. The model provides a useful mean for the practical approximation of the gap sensitivity of pressure losses.

Graphic abstract

Appendices

Appendix 1: Entropy creation rate for the mean flow

The first objective of this section is to clarify the definition of the entropy production rate for the mean flow. The second objective is to demonstrate that this entropy production rate can be estimated using velocity measurements from LDV. This work generalizes the analysis carried out by Denton (1993) on the relation between entropy increment and viscous forces. The following assumptions are considered :

• 1. The fluid is a perfect gas: $P=\rho R T$

• 2. The flow is in a steady state and incompressible: ${\text {div}}(\mathbf {u})=0$ and $M\ll 1$ (for the present case $M=0 .1$)

• 3. Variations of total pressure and temperature are small, i.e., $|P_\mathrm{{t}}-P_{\mathrm{{t}}\infty }|\ll P_{\mathrm{{t}}\infty }$ and $|T_\mathrm{{t}}-T_{\mathrm{{t}}\infty }|\ll T_{\mathrm{{t}}\infty }$ (in the present experiment $|P_\mathrm{{t}}-P_{\mathrm{{t}}\infty }|< 10^{-2}\times P_{\mathrm{{t}}\infty }$)

• 4. Heat transfers are neglected.

Starting with Gibbs–Duhem relation, one can relate entropy variation to total enthalpy and pressure for the instantaneous field following

$$\begin{aligned} T_\mathrm{{t}} {\hbox {d}}s = {\hbox {d}}h_\mathrm{{t}} - R T_\mathrm{{t}}\frac{{\hbox {d}}P_\mathrm{{t}}}{P_\mathrm{{t}}}. \end{aligned}$$

(28)

Assuming that $|P_\mathrm{{t}}-P_{\mathrm{{t}}\infty }|\ll P_{\mathrm{{t}}\infty }$ and $|T_\mathrm{{t}}-T_{\mathrm{{t}}\infty }|\ll T_{\mathrm{{t}}\infty }$ we have

$$\begin{aligned} T_{\mathrm{{t}}\infty } {\hbox {d}}s = {\hbox {d}}h_\mathrm{{t}} - \underbrace{\frac{R T_{\mathrm{{t}}\infty }}{P_{\mathrm{{t}}\infty }}}_{1/\rho _{\mathrm{{t}}\infty }}{\hbox {d}}P_\mathrm{{t}}. \end{aligned}$$

(29)

Insofar as the Mach number is small ($M=0.1$), we assume that $\rho \approx \rho _{\mathrm{{t}}\infty }$, which leads to

$$\begin{aligned} T_{\mathrm{{t}}\infty }\frac{{\hbox {d}} s}{\mathrm{{d}} t} = \frac{{\hbox {d}} h_t}{\mathrm{{d}} t} - \frac{1}{\rho } \frac{{\hbox {d}} P_\mathrm{{t}}}{\mathrm{{d}} t} \quad {\text {where}} \quad \frac{{\hbox {d}} \bullet }{\mathrm{{d}} t}=\dot{\bullet }=\frac{\partial \bullet }{\partial t} + u_i \frac{\partial \bullet }{\partial x_i}. \end{aligned}$$

(30)

Then, the Reynolds decomposition is applied to Eq. (30), considering the assumptions of a steady and incompressible flow, which leads to

$$\begin{aligned} T_{\mathrm{{t}}\infty }\,\dot{s}_\mathrm{{m}} =\langle u_i \rangle \frac{\partial }{\partial x_i} \langle h_t \rangle - \frac{1}{\rho } \langle u_i \rangle \frac{\partial }{\partial x_i} \langle P_\mathrm{{t}} \rangle , \end{aligned}$$

(31)

with $\langle \bullet \rangle$ the Reynolds average and with $\dot{s}_\mathrm{{m}}$ defined as

$$\begin{aligned} \begin{aligned} \dot{s}_\mathrm{{m}} =& \langle u_i \rangle \frac{\partial }{\partial x_i} \langle s \rangle +\frac{\partial }{\partial x_i} \langle u_i's' \rangle \\&- \frac{1}{T_{\mathrm{{t}}\infty }}\frac{\partial }{\partial x_i} \langle u_i'e' \rangle . \end{aligned} \end{aligned}$$

(32)

In (33), $e'$ represents the fluctuation of internal energy per unit of mass. This quantity $\dot{s}_\mathrm{{m}}$ can be understood as the equivalent rate of entropy production for the mean flow, which is not the same as the mean entropy production rate $\dot{s}$. From now on, the symbol $<\bullet>$ is dropped for mean flow quantities in order to reduce the amount of notation. Equation (31) thus becomes

$$\begin{aligned} T_{\mathrm{{t}}\infty }\,\dot{s}_\mathrm{{m}} = u_i\frac{\partial h_t}{\partial x_i} - \frac{1}{\rho }u_i\frac{\partial P_\mathrm{{t}}}{\partial x_i}. \end{aligned}$$

(33)

To replace the right-hand side of Eq. (33), let us consider the RANS energy equation

$$\begin{aligned} \begin{aligned} u_j \frac{\partial h_t}{\partial x_j}=&\left( 2 \nu S_{ij} + \tau _{ij}\right) \frac{\partial u_i}{\partial x_j} + u_i\frac{\partial }{\partial x_j}\left( 2 \nu S_{ij} + \tau _{ij}\right) \\&- \frac{1}{\rho }\frac{\partial q_j}{\partial x_j}, \end{aligned} \end{aligned}$$

(34)

and the RANS mean flow kinetic energy equation

$$\begin{aligned} \frac{1}{\rho } u_i\frac{\partial P_\mathrm{{t}}}{\partial x_i} = u_i\frac{\partial }{\partial x_j}\left( 2\nu S_{ij} + \tau _{ij} \right) , \end{aligned}$$

(35)

with $S_{ij}$ the strain rate tensor, $\tau _{ij}=-<u_i'u_j'>$ the Reynolds stress tensor and $q_j$ the heat flux. Inserting relations (34) and (35) in Eq. (33) leads to

$$\begin{aligned} T_{\mathrm{{t}}\infty }\dot{s}_\mathrm{{m}} = \left( 2 \nu S_{ij} + \tau _{ij}\right) \frac{\partial u_i}{\partial x_j} - \frac{1}{\rho }\frac{\partial q_j}{\partial x_j}, \end{aligned}$$

(36)

which is similar to the relation obtained by Chassaing (2010). This relation can also be written as follows

$$\begin{aligned} T_{\mathrm{{t}}\infty }\dot{s}_\mathrm{{m}} = 2 \nu S_{ij}S_{ij} + P_\mathrm{{k}} - \frac{1}{\rho }\frac{\partial q_j}{\partial x_j}, \end{aligned}$$

(37)

with

$$\begin{aligned} P_\mathrm{{k}}=\tau _{ij}\frac{\partial u_i}{\partial x_j}, \end{aligned}$$

(38)

the production of turbulent kinetic energy. Equation (37) shows that the equivalent mean flow entropy production rate $\dot{s}_\mathrm{{m}}$ corresponds to the sum of viscous dissipation ($2\nu S_{ij} S_{ij}$), production of turbulent kinetic energy ($P_\mathrm{{k}}$) and heat power (${\text {div}}(\mathbf {q})$). Upon assuming that the heat power is negligible, Eq. (37) yields

$$\begin{aligned} \dot{s}_\mathrm{{m}} = \frac{1}{T_{\mathrm{{t}}\infty }}\left( 2 \nu S_{ij}S_{ij} + P_\mathrm{{k}}\right) . \end{aligned}$$

(39)

This relation only depends on mean flow velocity gradients and on the Reynolds stress tensor; therefore, it is possible to compute $\dot{s}_\mathrm{{m}}$ from LDV measurements.

Appendix 2: Averaged total pressure losses and entropy creation rate

The objective of this section is to demonstrate the equivalence between mass-flow average total pressure losses $C_\mathrm{{Pt}}^\mathrm{{m}}$ and the volume integrated rate of entropy production $\rho \dot{s}_\mathrm{{m}}$ in a streamtube. The start is relation (33), which, taking into account that the flow is incompressible, leads to

$$\begin{aligned} T_{\mathrm{{t}}\infty } \rho \dot{s}_\mathrm{{m}}= {\text {div}}(\mathbf {u}. \rho h_t) - {\text {div}}(\mathbf {u}.P_\mathrm{{t}}). \end{aligned}$$

(40)

This relation is integrated over the streamtube $\Sigma$ composed of an inlet boundary $S_\mathrm{{in}}$, an outlet boundary $S_{0}$ and a lateral boundary $S_{lat}$. The inlet boundary is taken upstream of the blade where flow and thermodynamic conditions are homogeneous. The lateral boundary $S_{lat}$ is characterized by $\mathbf {u}.\mathbf {n}=0$, where $\mathbf {n}$ is the local normal vector. Then, the divergence theorem applied to the integral of (40) yields

$$\begin{aligned} \begin{aligned} T_{\mathrm{{t}}\infty } \iiint _\Sigma \rho \dot{s}_\mathrm{{m}} \, {\hbox {d}}V&= \iint _{_{S_\mathrm{{in}}}} P_{\mathrm{{t}}\infty }U_\infty (\mathbf {e}_x.\mathbf {n}) {\hbox {d}}S\\&\quad - \iint _{_{S_{0}}} P_\mathrm{{t}} (\mathbf {u}.\mathbf {n}) {\hbox {d}}S + \Delta H_t , \end{aligned} \end{aligned}$$

(41)

with

$$\begin{aligned} \Delta H_t=\iint _{_{S_{0}}} \rho h_{t} (\mathbf {u}.\mathbf {n}) \,{\hbox {d}}S -\iint _{_{S_\mathrm{{in}}}} \rho h_{\mathrm{{t}}\infty }U_\infty (\mathbf {e}_x.\mathbf {n}) \,{\hbox {d}}S. \end{aligned}$$

(42)

In the present study, we consider a fixed setup, i.e., no mechanical power is exchanged. Therefore, assuming no heat transfer at the boundaries of the streamtube $\Sigma$, we have $\Delta H_t=0$. Moreover, mass-flow being conserved between $S_\mathrm{{in}}$ and $S_{0}$, Eq. (41) becomes

$$\begin{aligned} T_{\mathrm{{t}}\infty } \iiint _\Sigma \rho \dot{s}_\mathrm{{m}} \, {\hbox {d}}V = \iint _{_{S_{0}}} (P_{\mathrm{{t}}\infty }-P_\mathrm{{t}})(\mathbf {u}.\mathbf {n}) {\hbox {d}}S . \end{aligned}$$

(43)

The mass-flow averaged total pressure losses $C_\mathrm{{Pt}}^\mathrm{{m}}$ is then introduced in Eq. (43), with

$$\begin{aligned} C_\mathrm{{Pt}}^\mathrm{{m}}= \frac{1}{\dot{m}}\iint _{S_{_{0}}} C_{P_\mathrm{{t}}}\,\rho \, (\mathbf {u}.\mathbf {n}) \,{\hbox {d}}S, \end{aligned}$$

(44)

where $\dot{m}$ represents the mass flow through the streamtube $\Sigma$, and with

$$\begin{aligned} C_\mathrm{{Pt}}=\frac{P_{\mathrm{{t}}\infty }-P_\mathrm{{t}}}{\frac{1}{2}\rho U_\infty ^2}, \end{aligned}$$

(45)

leading to

$$\begin{aligned} \iiint _\Sigma \rho \dot{s}_\mathrm{{m}} \, {\hbox {d}}V = \frac{1}{2} \frac{U_\infty ^2}{T_{\mathrm{{t}}\infty }} \,\dot{m} \,C_\mathrm{{Pt}}^\mathrm{{m}}. \end{aligned}$$

(46)

The average entropy increase per unit of mass $\mathcal {L}$, defined as

$$\begin{aligned} \mathcal {L}=\frac{1}{\dot{m}}\iiint _{\Sigma } \rho \dot{s}_\mathrm{{m}}\,{\hbox {d}}V \quad , \end{aligned}$$

(47)

is introduced in relation (46), which gives

$$\begin{aligned} \mathcal {L}= \frac{1}{2}\frac{U_\infty ^2}{T_{\mathrm{{t}}\infty }} C_\mathrm{{Pt}}^\mathrm{{m}}. \end{aligned}$$

(48)

Relation (48) states the equivalence between the mass-flow averaged total pressure losses and the average entropy increase per unit of mass.

Appendix 3: Steady axisymmetric vortex analysis

This section establishes a relation between the vortex loss coefficient $C_\mathrm{{vortex}}$, introduced in Eq. (13), and the circulation of the tip-leakage vortex. In a cylindrical reference frame $(x,r,\theta )$, $C_\mathrm{{vortex}}$ is defined by

$$\begin{aligned} C_\mathrm{{vortex}}=\frac{P_{\mathrm{{s}}\infty }-P_\mathrm{{s}}}{\frac{1}{2}\rho U_\infty ^2}-\frac{u_\theta ^2}{U_\infty ^2} \end{aligned}$$

(49)

The radial velocity $u_r$ is taken equal to zero. An axisymmetric steady inviscid flow is in radial equilibrium, that is

$$\begin{aligned} \frac{\partial P_\mathrm{{s}}}{\partial r} = \rho \frac{u_\theta ^2}{r} \end{aligned}$$

(50)

Upon introducing this relation in (49) leads to

$$\begin{aligned} C_\mathrm{{vortex}}= \frac{2}{U_\infty ^2} \int _r^\infty \frac{u_\theta ^2}{\xi } {\hbox {d}}\xi -\frac{u_\theta ^2}{U_\infty ^2} \quad . \end{aligned}$$

(51)

Let us now consider the Rankine vortex, defined as

$$\begin{aligned} u_\theta (r) = {\left\{ \begin{array}{ll} \frac{\Gamma }{2\pi a}\left( \frac{r}{a}\right) , \quad r<a \\ \\ \frac{\Gamma }{2\pi r},\quad r\ge a \end{array}\right. }, \end{aligned}$$

(52)

where a is the vortex core radius and $\Gamma$ is the vortex circulation. Based on this model, the coefficient $C_\mathrm{{vortex}}$ becomes

$$\begin{aligned} C_\mathrm{{vortex}}= {\left\{ \begin{array}{ll}\frac{2}{U_\infty ^2} \left( \frac{\Gamma }{2\pi a}\right) ^2\left[ 1-\left( \frac{r}{a}\right) ^2\right] , \quad r<a \\ ~ \\ 0,\quad r\ge a\end{array}\right. }. \end{aligned}$$

(53)

which shows that the $C_\mathrm{{vortex}}$ coefficient is maximum at the vortex center and is equal to zero outside the vortex core.

The surface average of $C_\mathrm{{vortex}}$ over a disk of radius R is defined as

$$\begin{aligned} C_\mathrm{{vortex}}^\mathrm{{s}}= \frac{1}{\pi R^2} \int _0^R C_\mathrm{{vortex}}(r)\cdot 2\pi r dr \end{aligned}$$

(54)

If the integration area includes the whole vortex core ($R>a$), one has

$$\begin{aligned} C_\mathrm{{vortex}}^\mathrm{{s}}= \frac{A}{S_0}\frac{\Gamma ^2}{U_\infty ^2}, \; S_0=\pi R^2, \; A=\frac{1}{4\pi },\; R > a \end{aligned}$$

(55)

Equation (55) shows that the surface average of $C_\mathrm{{vortex}}$ is proportional to the vortex circulation squared, divided by the integration area. This relation does not depend on the vortex core radius a. If $S_0$ and $\Gamma$ are normalized by the velocity $U_\infty$ and a reference length, relation (55) eventually becomes

$$\begin{aligned} C_\mathrm{{vortex}}^\mathrm{{s}}= \frac{A}{S_0}\Gamma ^{2}. \end{aligned}$$

(56)