2.2.1 Stationary solutions

Numerical results have been obtained for Reynolds gap numbers ranging between 600 and 30 000 to verify different trends with $Re_{s}$ . The product of the Eckert number and the dimensionless radial distance based on the gap, $E_{c}\cdot \tilde{r}^{2}$ , appearing in (2.17) is the Eckert number based on the radius:

Figure 4 shows the numerical solution of the Batchelor’s flow for $Re_{s}=2000$ , $Pr=0.71$ and $Ec_{r}=1$ . For gas turbine applications, whose working fluid is air, the Prandtl number is fairly constant in a large temperature range, and therefore its variation lacks interest and has not been pursued in this work.

Figure 4.

Dimensionless velocity components (a) and temperature field components $\tilde{T}_{0}$ and $\tilde{T}_{2}$ (b) at $Re_{s}=2000$ , $Pr=0.71$ , $Ec_{r}=1$ and $\tilde{r}=5$ .

Figure 2 sketches the flow pattern within two infinite disks. Fluid is pumped outwards by the rotor motion through a thin layer, sometimes referred to in the literature as von Kármán’s boundary layer. This thin region is fed by a central part, where viscous effects are negligible, which is termed here the core region. The fluid in this core region, to a first approximation, rotates as a rigid body with constant angular velocity and null radial velocity. There is a second thin region in the stator, generally termed Bödewadt’s boundary layer (Bödewadt Reference Bödewadt1940), with fluid moving inwards, to exactly compensate for the fluid expelled out by the rotor, finally enforcing the no-slip boundary condition at the wall. The relative thickness of these boundary layers with respect to the disk spacing is obtained by balancing the convective and viscous terms in the momentum equation, $\unicode[STIX]{x1D6FF}_{c}/s\sim Re_{s}^{-1/2}.$

Figure 4(a) also displays the temperature profile for the same case, where the three regions previously described for the velocity distribution can be noticed. The thermal boundary-layer thickness, $\unicode[STIX]{x1D6FF}_{T}$ , is of order $\unicode[STIX]{x1D6FF}_{T}/s\sim (Re_{s}Pr)^{-1/2}$ , in our case, since $Pr\simeq 1$ , both boundary-layer thicknesses are of the same order of magnitude, $\unicode[STIX]{x1D6FF}_{T}\sim \unicode[STIX]{x1D6FF}_{c}$ . It can be seen that the dimensionless temperature and circumferential velocity profile are similar, which is a clear indication that the Reynolds analogy is fulfilled.

Figure 4(b) shows the temperature components $\tilde{T}_{0}$ and $\tilde{T}_{2}$ for the same case. According to the component $\tilde{T}_{0}$ , the rotor can only heat a very thin region, of the order of $\unicode[STIX]{x1D6FF}/s\sim Re_{s}^{-1/2}$ , close to the disk. It can be seen that $\tilde{T}_{0}\simeq 0$ far away from the rotor. This is only possible because of the axial convection velocity, $H$ , from the stator to the rotor, which prevents the diffusion of the rotor heat into the core region. The quadratic component of the temperature, $\tilde{T}_{2}$ , is then the single mechanism to change the temperature of the core region by axial convection from Bödewadt’s boundary layer at the stator disk. As shown above for the previous assumptions, $\tilde{T}_{2}\sim E_{c}$ and $\hat{T}_{2}=\tilde{T}_{2}/Ec\sim 1$ , so if there is no dissipation $(Ec=0)$ , then $\tilde{T}_{2}\simeq 0$ and $\tilde{T}_{core}\simeq 0$ . It can be concluded that $\tilde{T}_{2,core}$ is created by a source in the immediate vicinity of the stator disk associated with the viscous dissipation, term $Ec/Re_{s}(G^{\prime 2}+F^{\prime 2})$ in (2.15). Temperature in the core region is then a function of the Eckert number based on the radius, $\tilde{T}_{core}\sim Ec_{r}$ .

Figure 5.

Renormalised non-dimensional shear stress ( $\tilde{\unicode[STIX]{x1D70E}}/Re_{s}^{1/2}$ ).

Angular momentum balance. The objective of this and the following section is to link the differential solutions with an integrated approach to the problem, which is frequently used in zero-dimensional models (Gärtner Reference Gärtner1998; Chew & Hills Reference Chew and Hills2013; Sun et al. Reference Sun, Amirante, Chew and Hills2016). Moreover, further insight can be gained by looking at the overall angular momentum and energy balance. The continuity equation, (2.6a ), and the momentum equation in the azimuthal direction, equation (2.6c ), can be combined to obtain the conservative form of (2.6c ),

Integrating (2.26) over a control volume defined from the wall to a distance well beyond the edge of the boundary layer, $\unicode[STIX]{x1D6FF}_{K}/s\gg Re_{s}^{-1/2}$ in the core region. The conservation of angular momentum in the overall domain can be separated into two independent control volumes, as shown in figure 5. Making use of the fact that in the core region $F_{core}=G_{core}^{\prime }=0,$ and at the wall $H_{wall}=0,$ the integral momentum equation becomes

The dimensionless shear stress at the walls can be expressed as

where $a_{\unicode[STIX]{x1D703}}$ and $b_{\unicode[STIX]{x1D703}}$ represent the universal non-dimensional shear stress in the rotor and stator, respectively. In the same line $f_{\unicode[STIX]{x1D703}}=d_{\unicode[STIX]{x1D703}}=H_{core}G_{core},$ and $c_{\unicode[STIX]{x1D703}}$ is defined as

in the rotor. The equivalent integral in the stator defines $e_{\unicode[STIX]{x1D703}}$ .

Table 1.

Constant values for the non-dimensional conservation of angular momentum.

Figure 5 outlines the tangential momentum equilibrium in a rotor–stator cavity identifying the terms of (2.27), whereas table 1 provides the actual values of the constants used in this figure. The tangential momentum generated by rotation of the rotor, $a_{\unicode[STIX]{x1D703}}$ , is convected outwards, $c_{\unicode[STIX]{x1D703}}$ , together with the azimuthal momentum convected in the core from the stator to the rotor, $d_{\unicode[STIX]{x1D703}}$ . This actually means that $c_{\unicode[STIX]{x1D703}}=a_{\unicode[STIX]{x1D703}}+d_{\unicode[STIX]{x1D703}}$ . Analogously, in the stator, momentum is radially convected inwards, $e_{\unicode[STIX]{x1D703}}$ , convected axially to the rotor, $d_{\unicode[STIX]{x1D703}}$ , and transmitted partially to the stator, $b_{\unicode[STIX]{x1D703}}$ . Momentum balances dictate that $e_{\unicode[STIX]{x1D703}}=b_{\unicode[STIX]{x1D703}}+d_{\unicode[STIX]{x1D703}}$ . It is interesting to note that since $a_{\unicode[STIX]{x1D703}}\neq b_{\unicode[STIX]{x1D703}}$ , the torque of the rotor is not fully transmitted to the stator.

Energy balance. Analogously to the azimuthal momentum equation, the continuity equation (2.6a ) can be combined with the energy equations (2.14) and (2.15), to obtain the corresponding energy equations in conservative form:

Again, integrating (2.30) over a control volume defined from the wall to a distance well beyond the edge of the thermal boundary layer, $\unicode[STIX]{x1D6FF}_{T}/s\gg Pr^{-1/2}Re_{s}^{-1/2}$ in the core region. The conservation of energy in the whole domain can be split into two independent control volumes. Making use of the fact that, in the core region $F_{core}=T_{0,core}=T_{0,core}^{\prime }=T_{2,core}^{\prime }=0,$ and at the wall $H_{wall}=0,$ the two integral energy equations become

Figure 6 sketches (2.31) in graphical form. Heat flux is non-dimensionalised using $q_{ref}=k(T_{R}-T_{S})/s$ and renormalised with $Re_{s}^{1/2}$ to absorb the dependence on the Reynolds number. The non-dimensional heat fluxes at the rotor and stator are then

where the explicit dependence on $Re_{s}^{1/2}$ can be seen. This definition is consistent with the classical definition of the Nusselt number based on the gap. All the terms affected by the $Ec_{r}$ have their origin in the quadratic terms of the energy equation. The rest of the terms of table 2 are defined as

Figure 6.

Renormalised non-dimensional heat flux balance ( $qs/k(T_{R}-T_{s})Re_{s}^{1/2}$ ).

Table 2.

Constant values for the non-dimensional energy budget for $Pr=0.71$ .

The rest of the constants in table 2, $c_{E}$ , $i_{E}$ , $h_{E}$ and $g_{E}$ , are the corresponding parts in the stator of $b_{E}$ , $d_{E}$ , $f_{E}$ and $e_{E}$ , respectively. As can be seen in figure 6, viscous dissipation has a determinant effect on the temperature near the rotor wall. In addition, we can estimate the value of the temperature at the core region using the energy balance previously studied. Assuming the conservation of mass flow over the whole domain, the dimensionless axial velocity at the core region can be estimated as

keeping in mind that $H_{B\ddot{o} \mathit{dewadt}}=1.35$ (Bödewadt Reference Bödewadt1940). On the other hand, using the numerical results of the energy balance shown in figure 6, we can derive $d_{E},$ and making use of the expression of the axial convection term in (2.33), we can estimate $\tilde{T}_{core}$ as a function of the dimensionless axial velocity at the core region as follows, using the constant values given in table 2:

Substituting (2.34) into (2.35) we can obtain an estimate for $\tilde{T}_{core}$ as a function of the swirl factor and the Eckert number:

Note that the radial variation of the temperature, $\hat{T}_{2}$ , is not controlled by the temperature difference between the rotor and the stator, since actually $T_{2R}-T_{2S}=0$ . The temperature in the core, $\hat{T}_{core}$ , is controlled by viscous heating in the stator and the subsequent convection from the stator to the rotor; windage heating contributes to increasing the temperature at the core region. The increment of temperature due to the dissipation can be estimated as

2.2.2 Transient solutions

Instantaneous spin-up. A previous study by Owen, Pincombe & Rogers (Reference Owen, Pincombe and Rogers1985) demonstrated that the steady solutions obtained with the present formulation were able to reproduce experimental results with reasonable agreement. In the present work, emphasis is placed on the transient evolution of a still fluid when the rotor is impulsively started. The same set of dimensionless parameters used in the stationary case has been kept. Although the initial and final states of the fluid are relatively simple and have been well studied in the past, the transient state is non-trivial because it involves the time-dependent evolution of a coupled convective and diffusive mechanical system.

Let us consider again the dimensionless circumferential momentum equation:

In the core region, $v_{\unicode[STIX]{x1D703}}\sim \unicode[STIX]{x1D6FA}r$ , and therefore $G\sim \mathit{O}(1)$ , the radial velocity is very small, $v_{r}\ll \unicode[STIX]{x1D6FA}r$ . Actually, in accordance with the dimensionless continuity equation, equation (2.6a ), $F\sim Re_{s}^{-1/2}$ , and therefore both convective terms of (2.38) (terms [2] and [3]) are of order $Re_{s}^{-1/2}$ , and then much larger than the diffusive term [4], which is of order $Re^{-1}$ . The time derivative is of the same order as the convective terms [1]–[2] or [3] when $\unicode[STIX]{x1D70F}_{c}=\unicode[STIX]{x1D70F}_{r}\sim Re_{s}^{1/2}$ . Physically, this is the time needed for a fluid particle to move from the stator to the rotor, $\unicode[STIX]{x1D70F}_{r}.$ This can be noticed by realising that the characteristic velocity in the axial direction is $v_{zc}\sim (\unicode[STIX]{x1D708}\unicode[STIX]{x1D6FA})^{1/2},$ and therefore

The dimensionless diffusive time, $\unicode[STIX]{x1D70F}_{d}$ , corresponds to the time taken by the diffusion to transport angular momentum in the axial direction. This would be the characteristic time if $H$ , i.e. the axial velocity, were strictly zero. In this case, the balance between the temporal derivatives and the viscous terms (term [4] in (2.38)) would yield $\unicode[STIX]{x1D70F}_{c}=\unicode[STIX]{x1D70F}_{d}\sim Re_{s}$ . This conclusion can be directly obtained by non-dimesionalising the diffusion time

There is a third characteristic time, which is the time taken by the diffusion to form the disk boundary layer, $t_{BL}\sim \unicode[STIX]{x1D6FF}_{c}^{2}/\unicode[STIX]{x1D708}=\unicode[STIX]{x1D6FA}^{-1}$ . This was the time taken to non-dimesionalise the equations, and hence, $\unicode[STIX]{x1D70F}_{BL}\sim \mathit{O}(1)$ .

Figure 7.

Dimensionless velocity components at the midpoint of the core region ( $\unicode[STIX]{x1D709}=0.5$ ) as a function of dimensionless time scaled with the diffusive time at $Re_{s}=2000$ for a disk impulsively started from rest (b). Velocity fields at (a) $\unicode[STIX]{x1D70F}=0.2Re_{s}$ (stage II) and (c) $\unicode[STIX]{x1D70F}=0.52Re_{s}$ (stage III).

The existence of three different characteristic times gives rise to three distinct phases of motion, namely, the rapid formation of the von Kármán boundary layer (stage I) where a quasi-stationary Stewartson-type of flow is created, a spinning-up (stage II) process by which the non-viscous core acquires its angular momentum, and finally the formation of a Batchelor-type flow field, once Bödewadt’s boundary layer is created at the stator and the swirl at the core is adjusted, balancing the rotor and stator boundary layers (stage III).

Figure 7 displays the time evolution of the axial and circumferential velocities for the impulsive start from rest ( $F_{0}=G_{0}=H_{0}=0$ at $\unicode[STIX]{x1D70F}=0$ ) of one of the disks of an infinite rotor–stator cavity at $Re_{s}=2000$ , computed with the numerical scheme previously outlined. At $\unicode[STIX]{x1D70F}=0$ , the rotor is impulsively set in motion, resulting in the very rapid formation of a Stewartson-type flow. This phase (stage I) can be seen as the development of a von Kármán boundary layer in a free disk, which is independent of $Re_{s}$ . The characteristic time of stage I is of order unity, $\unicode[STIX]{x1D70F}_{I}\sim \mathit{O}(1)$ . A von Kármán boundary layer is created in the rotating disk in which the azimuthal velocity decreases from the disk rotational velocity at the wall to the core velocity, which at this initial state is zero.

The subsequent stage II is highly sensitive to the details of the initial condition of the circumferential velocity. If $G_{0}$ is strictly zero at $\unicode[STIX]{x1D70F}=0$ , then the azimuthal velocity can only increase by diffusion from the disk wall, since the convective terms, [2] and [3], in (2.38) are strictly zero in the core region. The axial velocity, $H$ , constant and negative in the core region, is responsible for feeding the radial boundary layer created in the disk; however, it can only transport fluid with zero tangential momentum. This effectively means that the tangential momentum created by the disk is convected back to the disk; as a consequence the tangential momentum in the core region can only be increased by diffusion, and hence this stage lasts for a time $\unicode[STIX]{x1D70F}_{II}\sim Re_{s}$ , which is very long. This can be seen in figure 7(a) where a Stewartson-like velocity profile is observed in this stage.

The quasi-steady Stewartson-type flow rapidly changes after $\unicode[STIX]{x1D70F}\simeq 0.34Re_{s}$ : see figure 7(b). At this point, the tangential momentum, transported by diffusion from the disk, is radially convected inwards because the radial velocity in the core region of the Stewartson flow is negative. Since in the inviscid core region $rv_{\unicode[STIX]{x1D703}}$ is constant along the pathlines, a reduction in radius automatically implies an increase in the dimensionless circumferential velocity, $G$ . From a mathematical point of view the term [2] of (2.38) is no longer zero and triggers the exponential creation of circumferential momentum in a time $\unicode[STIX]{x1D70F}_{d}=\unicode[STIX]{x1D70F}_{II}\sim Re_{s}$ . Some researchers (van Eeten et al. Reference van Eeten, van der Schaaf, van Heijst and Schouten2013) describe this phenomenon as an instability of the Stewartson flow.

Once the core of the disk begins to rotate as a rigid body, a Bödewadt’s boundary layer is created in the stationary disk, in a time of the order of the diffusive time of the boundary layer, $\unicode[STIX]{x1D70F}_{BL}\sim \mathit{O}(1)$ . This can be seen in figure 7(c) where a Batchelor-type flow is depicted. However, during this stage III, the von Kármán and the Bödewadt boundary layers are not in equilibrium since they are not symmetric. For a given arbitrary swirl factor of the core, $\unicode[STIX]{x1D6FD}$ , firstly, the tangential momentum created in the core by both boundary layers is not in equilibrium, and secondly, the fluid ingested by the disk and egressed by the stationary disk does not match either. This global balance is established by convecting tangential momentum and mass from the stationary to the rotating disk in a time of the order of the residence time of the axial velocity of the core, $\unicode[STIX]{x1D70F}_{r}=\unicode[STIX]{x1D70F}_{III}\sim Re_{s}^{1/2}$ . The axial mass flow ingested by the disk during stage II of the process is smaller than that egressed by the stator once Bödewadt’s boundary is formed and the final steady state is reached (see figure 7 b). Since the flow is incompressible, this mismatch gives rise to a steep change in the axial velocity at $\unicode[STIX]{x1D70F}\simeq 0.52Re_{s}$ , in figure 7(c), and inertial waves are clearly seen in figure 7(b). We have tracked the duration of the spikes seen during the third phase of the transient stage, and we have also verified that its characteristic time scales as $\unicode[STIX]{x1D70F}_{c}\sim Re_{s}^{1/2}$ .

Figure 8.

Dimensionless time of stages II and III for different initial conditions.

We have solved the problem for different $Re_{s}$ and identified manually stages II and III. Figure 8 displays the duration of stage II (♦) and stage III (●) as a function of the Reynolds number. The slopes 1 and 0.5 are plotted to help the reader to draw their own conclusions. It is clearly seen that when the disk is impulsively started from rest, and the three components of the fluid velocity are strictly zero, the characteristic times of the stage II and III of the problem scale with $Re_{s}$ and $Re_{s}^{1/2}$ , respectively. These trends are more easily seen at the largest Reynolds numbers.

Stage II of the transient changes dramatically if the dimensionless circumferential velocity is initialised to a small value. Figure 8 displays the duration of stage II when $G$ is initialised to $10^{-4}$ (♢). It can be seen that, in this case, stage II is bypassed, and only the convective part of the process, which takes place in times of the order of $Re_{s}^{1/2}$ , can be seen. The starting process described when the fluid velocity is strictly zero at $\unicode[STIX]{x1D70F}=0$ , cannot be seen in practice in real engines, since frequently analyses are focused on the acceleration of the engine from idle to maximum take-off. The flow in the initial stage is already rotating ( $G_{0}\neq 0$ ), and hence the diffusive stage is bypassed. Idle rotational speed is approximately 10 % of maximum take-off rotational speed so that $G_{0}$ is actually much greater than $10^{-4}$ . Even if the disk is started from rest, there are always velocity perturbations associated either with the ventilation of the system or natural convection, if the engine is restarted, which are at least of the order of $10^{-3}$ . Hence our understanding is that the diffusive part of the process is always bypassed in realistic cases and it is not relevant.

It has been reported (Lopez et al. Reference Lopez, Marques, Rubio and Avila2009) that the spin-up and spin-down processes give rise to axisymmetric circular waves inside Bödewadt’s boundary layer. These radial variations cannot be captured by the similarity analysis presented here, which prescribes a linear variation of the velocity with the radius. Fortunately, these perturbations die out in very short times, $\unicode[STIX]{x1D70F}=\mathit{O}(1)$ . This actually means that this very short transient, although not seen in the solutions, hardly affects the global transient behaviour of the cavity, whose characteristic time is much longer.

Figure 9.

Transient evolution scaled with the diffusive time of the axial and circumferential dimensionless velocities (a) and dimensionless temperature (b) at the midpoint of the core region ( $\unicode[STIX]{x1D709}=0.5$ ) when the cavity is initially at rest ( $G_{0}=0$ at $\unicode[STIX]{x1D70F}=0$ ) and when it is uniformly initialised ( $G_{0}=10^{-4}$ at $\unicode[STIX]{x1D70F}=0$ ), at $Re_{s}=4000$ , $Pr=0.71$ , $Ec_{r}=1$ .

Figures 9(a) and 9(b) compare the evolution, for the same $Re_{s}$ , of the velocity and temperature fields in the core midpoint for two different initial conditions, $G_{0}=0$ and $G_{0}=10^{-4}$ . It is clearly seen that the final state is reached much earlier when the flow is started up with a fairly small perturbation than when the fluid is initially at rest. It can be concluded that, for $G_{0}=10^{-4}$ , the global dimensionless time evolution can be estimated as

Usually, heat transfer analyses of the start-up process of a gas turbine assume that the flow is quasi-stationary with respect to the solid to avoid computing a very long transient stage, which can become unaffordable from a computational point of view. However, we will show that there are analytical indications that demonstrate that this might not necessarily be the case.

Figure 10.

Transient evolution scaled with the diffusive time of the renormalised Nusselt number ( $Nu/Re_{s}^{1/2}$ ) at the stator and rotor with $G_{0}=10^{-4}$ , at $Re_{s}=4000$ , $Pr=0.71$ , $Ec_{r}=1$ .

The initial temperature of the fluid is assumed to be uniform and identical to that of the stator, $T_{i}(\unicode[STIX]{x1D709})=T_{S}$ . We now study the time-dependent evolution of the temperature field, when the disk is impulsively started and the fluid is at rest. As is shown in the time-dependent evolution of $Nu_{rotor}$ in figure 10, the thermal boundary layer at the rotor is nearly instantaneously created (stage I), in a dimensionless time of the order of unity, reaching a quasi-steady value of the heat flux at the rotor. Once the thermal boundary layer at the rotor has been created, $\widetilde{T}_{0}$ remains constant in the whole domain, and it is identical to that shown in figure 4(b) for the stationary case. The non-dimensional heat flux in the rotor, $Nu_{rotor}$ , is constant and equal to that of the steady solution, for $\unicode[STIX]{x1D70F}\gg 1$ (stages II and III).

Figure 11.

Dimensionless temperature field with $G_{0}=0$ (a) and $G_{0}=10^{-4}$ (b) at different dimensionless times for the case $Re_{s}=4000$ , $Pr=0.71$ , $Ec_{r}=1$ .

The time-dependent evolution of the temperature field separating the contribution of $\widetilde{T}_{0}$ and $\widetilde{T}_{2}$ , is studied in the following. The development of the temperature $\widetilde{T}_{0}$ can be studied from (2.14). Following the same argument as for the conservation of angular momentum in the azimuthal direction (2.6c ), it can be noticed that there is no term in (2.14) which can create an exponential growth of the temperature field and the contribution of $\tilde{T}_{0}$ to the temperature field is restricted to a narrow region in the vicinity of the rotor. In stage II, the temperature at the core region, which is solely due to the contribution of $\tilde{T}_{2}$ , increases due to the heating associated with the viscous dissipation term $Ec/Re_{s}(G^{\prime 2}+F^{\prime 2})$ in (2.15). The linear profile of $F$ gives rise to a constant source of heat. This can be seen in figure 11(a), where the instantaneous distribution of the temperature at $\unicode[STIX]{x1D70F}\simeq 0.5Re_{s}$ exhibits a linear profile due to the heating that a fluid particle undergoes while moving from the stator to the rotor. The transition of the temperature field from stage II to stage III is due to the appearance of a radial convection of temperature, represented by the term $2F\tilde{T}_{2}$ , in a dimensionless time of $\unicode[STIX]{x1D70F}\sim Re_{s}$ . This radial convection gives rise to the creation of Bödewadt’s boundary layer with its associated dissipation and heat source. The enthalpy created in the stator region gives rise to a step change of temperature from $\unicode[STIX]{x1D70F}\simeq 0.5Re_{s}$ to $\unicode[STIX]{x1D70F}\simeq 0.6Re_{s}$ , see figure 11(a). Finally the steady state is obtained approximately at $\unicode[STIX]{x1D70F}\simeq 0.8Re_{s},$ where the thermal energy created in the stator is transported first to the rotor and then radially in the rotor boundary layer.

If we now consider the case for which at $\unicode[STIX]{x1D70F}=0$ , $G_{0}=10^{-4}$ , analogously to what happened in the velocity field, the diffusive (stage II) is bypassed. As shown in figure 11(b) at $\unicode[STIX]{x1D70F}\simeq 0.32Re_{s}$ , the distribution of temperature along the axial direction is not constant, so when the Bödewadt boundary layer is created, the global balance is established by axial and tangential convection, until it reaches the steady solution, as shown at $\unicode[STIX]{x1D70F}\simeq 0.6Re_{s}$ in figure 11(b). In this case, simultaneously with the creation of the stator boundary layer, the heat created in the stator region is transported to the disk; since the heating in the stator increases with time, the slope of the temperature profile at $\unicode[STIX]{x1D70F}=0.32Re_{s}$ is negative.

Concerning the transient development of the thermal field, if we define the local Nusselt number, $Nu_{l}$ , using the instantaneous bulk temperature of the core region, then

If we assume that $Nu_{rotor}$ remains constant along time evolution (see figure 10), since the temperature in the core varies with time (see figure 9), the local Nusselt number previously defined increases with time.

Instantaneous spin-down. The transient evolution of the velocity and temperature in the core region of an infinite cavity between two disks, when initially one disk is rotating and the other is at rest, and the rotor is impulsively stopped, is now studied. The first stage of the process is the very rapid destruction of the von Kármán boundary layer of the rotor and the creation of a new Bödewadt boundary layer (stage I) in dimensionless times of order unity, $\unicode[STIX]{x1D70F}_{I}\sim \mathit{O}(1)$ . The azimuthal velocity profile increases from zero at the rotor wall (now at rest) to the core velocity, which is the one corresponding to the steady state before the spin-down.

A sketch of this initial stage is shown in figure 12(a). Once the circumferential velocity in the core of the cavity begins to decrease due to viscous effects (stage II), the Bödewadt boundary layer in the originally stationary disk is adapted to the new conditions outside the boundary layer. The axial velocity, positive in the left disk and negative on the right disk, is instantaneously changed from a negative value to a positive, which means that the convective axial component transports the temperature at the rotor to the core region until the solution reaches steady thermal equilibrium where $\widetilde{T}_{core}=0.5$ .

Figure 12.

Dimensionless axial, circumferential velocities and temperature at the midpoint of the core region ( $\unicode[STIX]{x1D709}=0.5$ ) as a function of dimensionless time scaled with the diffusive time at $Re_{s}=4000$ for a spin-down regime where the rotating disk is suddenly stopped (b). Velocity fields at (a) $\unicode[STIX]{x1D70F}=0.02Re_{s}$ (stage II) and (c)  $\unicode[STIX]{x1D70F}=0.1Re_{s}$ (stage III).

During this stage, the two Bödewadt boundary layers are not in equilibrium since, in general, they are not symmetric due to the fact that the swirling factor of the core, $\unicode[STIX]{x1D6FD}$ , is steadily decreasing with time. Both the tangential momentum and the axial mass flow are balanced by means of inertial waves which give rise to steep gradients in all the variables, especially in the axial velocity, as shown in figure 12. The global balance of momentum and mass is continuously updated by axial convection in a time of the order of that needed by the fluid particles to move from the stator to the core, $\unicode[STIX]{x1D70F}_{r}\sim Re_{s}^{1/2}$ , giving rise to oscillations in the axial velocity.

The subsequent stage III is of a diffusive nature and its characteristic time is of the order of $\unicode[STIX]{x1D70F}_{t}\sim Re_{s}$ . The fluid reaches the actual rest state, although at the end of stage II it can be considered that the fluid velocity is zero. The most noticeable macroscopic effect during this phase in the diffusion of the temperature field, until a linear temperature profiled is attained in the cavity. However, although hardly seen in figure 12, $G$ and $H$ also are smoothed out by diffusion. The characteristic time of this stage is $\unicode[STIX]{x1D70F}_{III}\sim Re_{s}$ , which is very long. This can be seen in figure 12(c).