Effects of System Rotation on Turbulence Structure : A Review Relevant to Turbomachinery Flows

Turbomachine rotor flows may be affected by system rotation in various ways. Coriolis and centrifugal forces are responsible for (i) modification of the structure of turbulence in boundary layers and free shear layers, (ii) the generation of secondary flows, and (iii) “buoyancy” currents in cases where density gradients occur. Turbulence modification involves reduction (stabilization) or increase (destabilization) of turbulent Reynolds stresses by Coriolis forces; effects which areof special importance for the understanding and prediction of flows in radial and mixed flow pump and compressor rotors. Stabilization/destabilization effects are discussed by a selective review of the basic research literature on flows in straight, radial, rotating channels and diffusers.


INTRODUCTION
In 1970 a symposium on "Fluid Mechanics, Acoustics and Design of Turbomachinery" was held a Pennsylvania State University where, in session II, the current state of knowledge on effects of system rotation on boundary layers in turbo- machine rotors was reviewed and discussed by the attendees, Johnston (1974).The fact that Coriolis forces cause significant secondary flows (crossflows from pressure to suction side) in the end- wall boundary layers of radial and mixed flow pump and compressor impellers had already been clearly established, but direct experimental evidence concerning Coriolis effects on turbulence and transition in boundary layers and free shear layers was new at the time of the symposium.Many of the issues discussed at the Penn State meeting had their origin in questions raised in the early 1960s by R. C. Dean Jr., see Dean (1968) where his observations concerning flow in radial flow compressor impellers are published.It is the objective of this paper to review our knowledge on these subjects, 27 years later.This brief, selective review is not aimed for the research community.
Rather, it is intended for the design engineer trying to understand and predict rotor flow phenomena that may at first glance seem unusual.
Tel.: (415)723-4024.Fax: (415)723-4548.E-mail: Johnston@vonkarman.stanford.edu.98 J.P. JOHNSTON   Rotor flow fields are most conveniently exam- ined in coordinates which rotate with the rotor at steady angular velocity, D, about the axis of the machine.The fluid velocity, U, measured with respect to rotating coordinates, obeys the conserva- tion of mass relation, Ot --+ v. (pu) 0, when the fluid density, p, is variable.However, when p is constant, Eq. (la) reduces to the simple continuity equation, V.U=0.(lb) Note that symbols in bold are vectors, and X7 is the vector gradient operator which when multiplied (scalar product) into a vector such as U produces the sum of spatial gradients of the vector, The dynamic equation of motion for a fluid particle, in steadily rotating coordinates, may be given by DU Dt particle acceleration 2r2 --7p 2U x D + 27 + + Fvis. (2)  2 p Coriolis / centrifugal + pressure force force force + viscous force There are four forces (per unit volume) which control the fluid's motion.The Coriolis and centrifugal forces are 'virtual' forces that result from the use of rotating coordinates whereas pressure and viscous forces are the real forces on a fluid particle in any coordinate system.When uX72(U) is substituted for Fvis, the incompressible, constant viscosity Navier-Stokes equations are obtained, where u #/p, is the kinematic viscosity.

2O FIGURE
Coriolis and centrifugal forces (per unit volume).
on a fluid particle.
The centrifugal force has a magnitude of [Q2r] and acts along the local radial direction as shown in Fig. 1.The figure also shows the direction and magnitude, [2UF sin(e)], of the Coriolisforce.This force is perpendicular to a plane formed by the U and D vectors, and is directed as shown, according to the right hand rule of the vector cross product.
The angle e ranges from 0 to r/2.Since e is close to r/2 in the case of radial flow rotors, it may be anticipated that Coriolis effects could be large in such cases.For flow in axial flow rotors where the radial component of U is very small Coriolis forces are also small and they point in the radial direction.
In the case of a constant density flow, the centrifugal force plays no independent role in the dynamics of motion as it may be combined with the pressure force term to form a single term, -g(p*/p) by the definition ofan 'effective pressure', p*=p+p(Zr2)/2.In general however, where density gradients, X7p, exist in the flow, the For Cartesian coordinates, x; (i= 1,2, or 3) where the vector's components are Ui, the dot (scalar) product gives the sum of deriviatives; 27.U =OUJOx=OU1/Oxl +OU2/Ox2 +OU3/Ox3.The cross (vector) product, 27 U represents the curl of the velocity field, and yields the vorticity (2 x fluid rotation) with respect to rotating coordinates.The vorticity in inertial space (stationary coordinates) is 2f* + 27 x U.In rotating Cartesian coordinates, the vorticity in the x direction is (OU2/Ox3-OU3/Ox2).EFFECTS OF SYSTEM ROTATION 99 centrifugal force is similar to a gravitational body force (neglected her'e) and thus may contribute to the flow's dynamics.
The Coriolis force has the interesting character- istic that it can't affect the energy of a flow directly.
Change of energy of a fluid particle is affected by an applied fo.rce, F, according to the work, F. dx, done on the particle when it is displaced through the distance dx.Since instantaneous displacement in rotating coordinates is along the U vector, and because U is perpendicular to the Coriolis force vector there can be no transfer of energy to/from the flow by the Coriolis force.This has the consequence that both the mean flow and the turbulence kinetic energy, k, cannot be directly produced or destroyed by Coriolis force.However, both the normal stress and the shear stress components of the turbulent Reynolds stress tensor, -uiuj, may be affected by this force.

Effects of Rotation
The effects of rotation may be illustrated using a very simple example of a rotor flow, the flow along a radial rectangular channel as depicted in Fig. 2. Typically one expects a core region of high speed flow with lower speed fluid in the end-wall and side-wall boundary layers as shown in the isometric view.An END VIEW, looking along the center line of the channel shows a shaded area representing regions in the boundary layers where the relative velocity, U, is low with respect to its value in the core.The Coriolis force is shown in the figure by the heavy arrow marked 2UFt.Its value will be largest in the core where U is at its maximum value.The core region Coriolis force will establish a pressure gradient between the two side walls causing high pressure, pp, near one (pressure) side wall and lower pressure, Ps, near the opposite (suction) side wall.To a degree of approximation the cross-stream pressure gradient in simple channel flow is (Pp -Ps) 2h p(2 Ua). (3) The pressure gradient established in the core will be felt inside the end-wall boundary layers, according to boundary layer theory.Because of the lower flow speed in the boundary layers they will develop lower Coriolis forces, and consequently fluid particles inside the end wall layers must be accelerated and move toward the suction side wall, in the direction indicated by the short arrows near the end walls.This cross-flow is called the secondary flow due to system rotation.
Coriolis driven secondary flows have different relative effects on rotor flows depending on many geometric and flow variables.In the case of simple rectangular channels, various investigations have examined rotational effects, and one finds that channel aspect ratio (AS=b/h) is an important parameter.Figures 3 and 4 are used.toillustrate the point.Two cases, low and high aspect ratio, are sketched in the first figure.They show that secondary streamlines established by the accelera- tion of the fluid in the end wall boundary layers Cartesian compact subscript notation is used in discussions of turbulence, xi (i 1,2, or 3) represents the x, y, or z directions, Ui represents U, V, or W, the time mean or ensemble average components of the velocity, ui represents u,v, or w, the velocity fluctuations about the mean.Ul N is a typical average of the fluctuations along (x-direction), and k 1/2(u--i-+ + u-5-) is the turbulence kinetic energy.-uiuj (i 1, 2, or 3, andj 1, 2, or 3) is the Reynolds stress tensor; it has six components, the three shear stresses where Cj and the three normal stresses where =j.Moore (1967).
leading (pressure) side-wall as AS increases, and vise versa on the trailing (suction) side-wall.Moore also varied the rotation rate to show that the effect increased as rotation rate increased.
A surprising feature of these results is the relatively large effects seen for the channel of largest aspect ratio, AS 7.3.In this case, the secondary flow in regions far from the end walls would be expected to cause a + or 2 percent change in side wall skin friction rather than the + 10 to 15 percent changes noted by Moore.These large increases or decreases are not secondary flow effects, but they result from destabilization (increase) or stabilization (decrease) of turbulent shear stresses in the side-wall boundary layers themselves.These effects are the main topic of this paper.
In addition to the (i) secondary flow effects and (ii) the turbulence stabilization effects in rotating systems, one needs to be aware that (iii) centrifugal forces can also contribute substantially to the flow.This force is particularly important when large density gradients exist, as might be the case for the flow inside blade cooling passages of gas turbine rotors.The centrifugal force is always radial and acts outward with respect to the axis of rotation, see Fig. 1.Consequently, it causes fluid forces which tend to drive low density fluid toward the axis of rotation and high density fluid away from the axis.Because of the similarity to the effects of gravity in a stationary case, the flows induced by centrifugal forces are sometimes designated as centrifugal buoyancy currents.This is a special topic which will not be discussed further in this paper.tend to cover the whole cross section of the low aspect ratio channel, but when AS is high, the secondary streamlines are concentrated near the end walls leaving the core region unaffected.
Moore (1967) experimented with turbulent boundary layer flow in rotating channels of differing aspect ratio and showed that secondary flows had large effects near the center line of symmetry when AS was low, compared to the effects when AS was high.Figure 4 shows a decrease of the skin friction coefficient, Cr, on the

Stability and Instability
These terms are used here to denote the tendency for a boundary layer or a free shear layer to experience increased or decreased levels of turbulence with respect to the 'natural' levels that would develop under non-rotating conditions.Nonrotating conditions are the basis for models used to estimate the turbulent Reynolds stresses, -uiuj.These models then provide the equations for closure of the set of mean flow equations, the Reynolds average Navier-Stokes (RANS) equa- tions.For example, Johnston and Eide (1976) applied empirical algebraic corrections to a mixing length method in order to model the effects of rotation on the turbulent shear stress, ---.Param- eters for expressing these effects were derived from knowledge concerning the effects of rotation on flow stability.
A boundary layer velocity profile U(y) on a rotor blade is illustrated in Fig. 5. Here, 2Uw, is the component of the Coriolis force along the y-axis which is defined to be perpendicular to the blade surface, co= f sin()cos() is the effective rate of rotation in regard to stabilization effects.For example, in radial flow rotors e r/2 and 0 so w f.If the direction of the force is positive along the y-axis (as drawn), then the boundary layer will be stabilized.Which means that a fluid eddy displaced along y in either the + y direction feels a restoring force which tends to return it to its original y location.Conversely, if the effective Coriolis force is in the reverse orientation, pointing toward the wall, displaced eddies feel forces that tend to increase their displacements from their original location, the situation is destabilizing.Stabilization effects were discussed and proper parameters for their description developed by Bradshaw (1969).His parameter, Ri, has become   widely used in all subsequent work.Ri, by analogy named the gradient Richardson number, permits one to describe the degree of stabilization at each y position in a velocity profile where the local velocity gradient is OU/Oy. -2w Ri-S(S + 1), where S-OU/Oy" (4)   The flow is locally stabilized when: Ri > 0, and it is destabilized when Ri < 0. The flow stability condi- tions are neutral (Ri --0) for two values of S, when S 0 and, under rotating conditions, when S -1.This latter condition is interesting in that the vorticity of the velocity profile with respect to inertial (non-rotating) coordinates, (2w-OU/Oy), is also zero for this neutrally stable condition.
In a radial or mixed flow pump or compressor impeller, according to these criteria, the boundary layers on the suction (trailing) sides of the blades will have Ri > 0 and hence they will be stabilized, but on the pressure (leading) sides flow will be destabilized by rotation.In the inviscid core region, should one exist, the flow is generally inertially irrotational and thus it is neutrally stable.The consequences of these effects, are that increased turbulence and mixing in destabilized (pressure side) boundary layers makes them less likely to separate, but the stabilized layers on the suction side become less turbulent and more prone to flow separation in regions of adverse pressure gradient.
Developments Since 1970   These are first outlined to give a brief overview on progress in the field over the past quarter century.For the most part, this review will concentrate on results obtained by experiment.However, some recently computed results obtained by direct numerical simulation (DNS) of the Navier-Stokes equations are also discussed.Turbulence modelling for use in RANS methods is not reviewed because such models must result from basic understanding which can only come from experiment and/or DNS.
Newtonian viscosity, #, must be introduced as a fluid property to properly represent viscous forces in eitherexperimental or DNS results.However, because the density, p, is constant for all the work reviewed here, the only significant fluid property is the kinematic viscosity, u #/p.Thus, for a system of given geometry one may characterize all dimensionless forces, flow rates, etc. in terms of two dimensionless parameters; a Reynolds number, Re--VA/u, and a Rotation number, Ro 2fA/V.
V and A are a fluid velocity scale and a length scale respectively.They must be specified according to the needs of the problem at hand.The Rotation number represents a ratio of Coriolis forces to inertia forces (particle accelerations).The magni- tude of the Rotation number, ]Ro] increases from zero as rotation increases, and thus is useful to scale the effects of rotation which, to a first approximation, are found to increase (destablization effect) or decrease (stabilization effect) linearly as Ro increases.
Because the effects of rotation are concentrated in sheared regions such as boundary layers, it is useful, in order to understand the physical processes, to take V-Ue, the relative velocity at the edge of a boundary layer, and A=5, the boundary layer thickness.Therefore, Re= Ue/u and Ro--2co(5/Ue are physically meaningful defini- tions of the two controlling parameters.By taking OUe/Oy U/6 as an estimate of the mean shear in a boundary layer, One obtains an estimate of the mean global stability parameters, S-Ro, and Ri Ro(Ro-1).Johnston (1974) showed that the magnitude of the boundary layer Rotation number is expected to be low (]Ro < 0.1) in turbomachines even though the effects of rotation are substantial at these low values.Finally, by combining esti- mates it is seen that the global stability of a rotor blade boundary layer is approximately equal to the Rotation number, i.e., Ri -Ro.This is a justifi- cation for using the Rotation number in turbulence modeling, e.g., see Johnston and Eide (1976).
Although some excellent data sets have been obtained by means of laser anemometry in real compressor and turbine rotors, these flow fields are too complex to allow one to access basic physical phenomena systematically, one at a time.Consequently, basic research investigations have gener- ally employed geometrically simple flows, a number of which are listed in Table I.The cases listed used air or water as the working fluid and were operated at ]Ro] < 0.2 and at Reynolds num- bers which caused boundary layers to be turbulent, conditions characteristic of many turbine and compressor rotor flows.Thirteen experiments and one DNS case utilized radial rotating channels, constant area ducts of rectangular cross section.Three experiments in this group concerned themselves with detailed measurements of mean velocity profiles and Reynolds stresses in the turbulent boundary layers, and two studies addressed the effects of stabilizing rotation on transition to turbulence.Four studies were conducted in two- dimensional, plane-walled diffusers.The diffuser flows simulated the effects of adverse pressure gradients along the blades of radial flow pump and compressor impellers where flow separation is often observed on the suction sides of the blades.Some radial, rotating pipe flows and conical diffuser flows are also listed in Table I because they may provide useful information in special applications, such as flow in turbine blade cooling ducts.However, the round cross section of a pipe makes it hard to disentangle stability effects from secondary flow effects.For completeness, two basic experiments on turbulent free shear layers in rotating systems are listed.Finally, the aspect ratios for the various channels and diffuser inlet planes are listed in the right hand column of Table I.
The inverse of the Rotation number, Ro-1, called the Rossby number, is generally used in geophysical fluid dynamics where very high values of Ro are often encountered.

FOUR SITUATIONS
The effects of rotation on stabilization are discussed by reference to four distinct situations using five cases from Table I (**), all chosen because stability effects dominate and the effects of secondary flows are negligible.These cases have high values of channel aspect ratio, AS _> 4. The situations are: (i) fully-developed flow in long rotating channels; (ii) turbulent boundary layers on the walls of a straight rotating channel; (iii) plane walled two-dimensional diffusers with rota- tion axis perpendicular to parallel end-walls, and (iv) back-step diffuser flow with separation at the step's sharp edge and reattachment downstream.This is an extensively studied case, for which the experiments of Johnston et al. (1972), and a direct numerical simulation (DNS) by Khristofferson and  Andersson (1993) comprise the useful data base on stabilization effects.For these fully developed flows the boundary layer thickness h, half the width of the channel.Also, the appropriate scaling Wl "S" Suction (Trailing) Side "P" Pressure (Leading) Side FIGURE 6 Diagrams illustrating four special situations.Note: S and P side-walls are inverted if 2 vector reverses direction.
velocity is the mean velocity in the channel, Um.
Therefore, the physically significant parameters are defined as: Re=hUm/v and Ro=2h/Um.The DNS was conducted at Re=2900 and 0 < ]Ro] < 0.5.The experimental results covered a wider range of Reynolds numbers but narrower ranges of Rotation numbers: from Re 5500 with 0 < IRo] < 0.2, up to Re 18,000 with 0 < ]Ro] < 0.1.
The flow visualization results from Johnston et al.
(1972) and Lezius and Johnston (1976) clearly showed how rotation changes the details of the turbulent flow on suction and pressure surfaces.
Results from the 1993 DNS study confirm almost all the experimental observations from the early 1970s.In summary, it is found that: (a) On the stabilized (suction) side increase of ]Ro] decreased turbulence activity continu- ously, at all Reynolds numbers.For Re 5500, when ]Ro] 0.2 the stabilized wall layer was so stable that the large energetic struc- tures, called roll cells (described below), which crossed over from the pressure side of the channel could excite only a rapidly damped transition response near the suction wall.At higher Re values, the effect was less dramatic, the flow in the stable side wall layers remained turbulent, but with reduced turbu- lent shear stress, --, and a drop in turbu- lence mixing as indicated by lowering of v, the component of turbulence normal to the wall, along y.
(b) On the unstable (pressure) side the opposite effect is seen, turbulent shear stress and mixing increased steadily as ]Ro] increased.
(c) In the unstable regions of the flow a particular instability phenomenon was observed, the development of large scale roll cells.Such cells are predicted by application of hydrodynamic stability theory for low Reynolds number, laminar, rotating channel flow, and the cells have some of the characteristics of Taylor- Hart (1971) and Lezius and Johnston (1976).However, in turbulent channel flows the cells appeared as a quasi periodic pattern of eruptions which moved away from and perpendicular to the pressure wall.The spanwise spacing of these eruptions was roughly equal to the channel half width, h.Between the outward, rather T-G vortex cells have weak secondary (y-z plane) circulations about axes along the mainflow direction.
turbulent eruptions, calmer fluid from the suction side of the channel returned slowly to the unstable region.An individual roll cell was defined as the combination of half an eruption and half a calm return flow.The cells were never stationary in space, but tended to continuoUsly breakup and reform in different locations over time.Compared to the short lived, smaller scale and very chaotic turbulence, the more organized roll cells had comparatively long time scales, and hence were thought of as quasi stationary events.
The consequence of these changes in flow struc- ture are seen by examination of a few quantitative measurements.Mean velocity profiles from both the experiments (a) and from the DNS results (b) are shown in Fig. 7.It is clear that changes in turbulence structure are having major effects on the mean velocity distributions, even for very small values of IRol.The straight lines drawn tangent to the central portions of each profile have slopes that correspond to the condition ofirrotational flow with respect to stationary coordinates.The zones where the data points lie on these lines are zones of neutral stability, according to the stability criteria set out above.As ]Rol increases, the neutrally stabile portion of a profile increase in length, a result that has been seen in other fully-developed turbulent flows where roll cell or Taylor-G6rtler vortices can develop.**Flow visualization suggests that the roll cells are the primary cause of the region of neutral stability; it is observed that the roll cell eruptions increased in strength as IRol increased.FIGURE 7 Mean velocity profiles for fully-developed turbulent channel flows.Notes: The stabilized side is at y/h=-1 and destabilized side at y/h + 1.In (b), U nondimensionalized using u, the global wall shear velocity (see reference).
Flow in a long channel of constant width with constant inner and outer wall curvatures, and flow in the gap between concentric cylinders where the inner cylinder rotates are primary examples.Figure 8 illustrates the effects of rotation on the wall skin friction stresses in terms of the wall friction velocities, u_=(-wall/p)5, on suction (c s) or pressure (a p) walls.The global average value, uT, used for normalization is based on the streamwise pressure gradient, -dp*/dx.Wall fric- tion rises continuously on the pressure, wall, but appears to level out at higher IRo] values.On the suction wall, at higher IRo wall friction drops to quite low levels.There is evidence in the flow visualization that damped laminar flow may exist in a thin layer near the suction surface at the highest IRol values for the lowest Reynolds numbers.The results from a recent large eddy simulation (LES) are also shown on Fig. 8, and except for values of IRol < 0.05, these results seem to be in good agreement with the rest.
and 8 and fairly good agreement has been attained when the levels of Ro are low.Kristofferson and Andersson (1993) and a more detailed analysis of the CFD results by Andersson and Kristofferson (1993), provide the best current information for successful turbulence modeling of turbulent boundary layer and channel flows with moderate to high system rotation strengths, Ro up to A: 0.5.
(ii) Turbulent Boundary Layers on the Walls of Straight Rotating Channels These were studied by two groups, Koyama et al. (1979, 1995) and Watmuff et al. (1985).The discussion concentrates on Watmuff's case as his experiment simulates conditions of a two-dimen- sional turbulent boundary layer with zero free stream pressure gradient.The pressure gradient was controlled by adjustment of the angle between channel side walls so that the free-stream velocity was constant from inlet to exit along the whole length of the side wall on which boundary layers were measured.Secondary flow effects from the end-wall cross flows were negligible in the central regions of the test surface because the boundary layers were thin.They grew, downstream of a trip, to a maximum thickness 25% of the channel width, 2h, and thus the effective aspect ratio was twice the actual ratio defined by the walls, AS-4.
Watmuff's air flow study reports hot wire (turbulence) and total head tube (mean velocity) profiles at five downstream stations for six different flow conditions.He used three free stream velocities (Ue 7.5, 10, and 15 m/s) and three rotation rates (stabilizing at t2=-60rpm, 0rpm, and destabi- lizing at + 60 rpm).The Reynolds numbers, where Re=Ue/u, and magnitudes of the Rotation numbers, where IRol=21fl/Ue, achieved maxi- mum values at the downstream station where 35mm: Remax 1.7 x 104 to 3.5 104 and ]Rolmax 0.06-0.03over the range of free stream speeds and rotation rates.These parametric values are approximately the same as those encountered in the boundary layers over the blades of radial flow compressors, as noted in Johnston (1974).
Many of the general conclusions drawn from the fully-developed rotating channel cases were seen in this case too, i.e., stabilizing rotation reduced turbulence and wall shear stress, and destabilizing rotation had the opposite effect.Also, the large scale roll cells were observed in the destabilized boundary layer.However, in Watmuff's cases the roll cells tended to be much steadier in space and time than seen previously in the rotating channel flows.Another important difference from earlier channel flow studies is that the roll cells did not interact with the boundary layer on the stabilized side, because a thick, inviscid core region separated the suction and pressure side boundary layers by at least 26 in all cases.
Some of the most important practical contribu- tions from the study concern the use of scaling parameters to describe the mean velocity profiles.
For example, in practical computations one is often limited in the density of mesh points that may be used inside thin boundary layer zones.Therefore, it is necessary to prescribe empirical wall functions in order to satisfy boundary conditions at a point close to a solid wall rather than apply no-slip conditions at the wall itself, y =0.Typically, this inner boundary point is at y/6 0.05-0.1,inside a zone of overlap between inner and outer layers.The overlap region often called the "universal log law" region, because, by use of wall layer scaling, tt the mean velocity profile in the overlap layer generally tends to follow the formula: U+ 1 ln(y+) + C + A U+. (5) For non-rotating, turbulent boundary layers, the constants are usually given as =0.41,C= 5.0, and the offset, A U+, is set to zero for smooth surfaces and low turbulence free stream flows.
Some mean velocity profiles are shown in Fig. 9 to illustrate the general effects of rotation on the profile shapes.The upward shift in the outer regions for the stabilized case is caused by reduc- tion of turbulent shear stress and vice versa for the destabilized case.The log region is very short for the stabilized case and quite a bit longer in the destabilized flow.Reduction in size of the logarithmic overlap zone is generally in indication of a trend toward lower turbulence and reduced mixing.
effects, Bradshaw (1969) and others have sugges- ted, on the basis ofstability arguments, that one may prescribe the offset using the simple linear relation AU + -/3 ,2coy where co the effective rotation rate of Fig. 5, is used here.co is negative for a suction (stable) side boundary layer and co is positive for a pressure (unstable) side layer.An empirical "constant" /3 4 + 2 fits the overlap zone in the channel flow data of Johnston et al. and Watmuff's unstable side profiles fairly well.However, Watmuff's stable side velocity profiles fit the linear offset relation very poorly, if at all.This observation, together with the wide range of possible values of/3 makes the A U + method highly uncertain and therefore not very useful as a Coriolis force correction to the log law.suggest a better empirical fit in the overlap zone tt Wall layer scaling uses u for velocity and u/u for length, so dimensionless velocity is U + U/u and dimensionless distance from the wall is y + yu-/u.where A U + ---=0 but and C are no longer held constant.Both suggest that and C should be expressed as functions of a Rotation number based on wall layer scaling, cou/u.For example, Watmuff et al. produced a formula, I--2.4+750(' u) (7) that fits the data quite well, as is seen in Fig. 10.
A similar result would be obtained for the variation of C with cou/u2.From a practical vantage, this method is probably the best idea to date for a wall function in rotating boundary layers.
(iii) Plane Walled Two-Dimensional Diffuser Flows with Rotation Axis Perpendicular to the Parallel End-Walls These approximate the flows seen in the bladed rotors of radial flow pumps and compressors.The simplest cases are the four experiments on straightwalled diffusers noted in Table I.Unfortunately, Fowler (1968) did not document boundary layer regions in detail, and, like Moore's (1973) rotating diffuser flow, Fowler's experiment was conducted with a low aspect ratio at the diffuser's inlet.
Because of the low aspect ratios, these two cases show substantial end-wall secondary flow effects which tend to confuse the interpretation of the data in regard to stabilization effects.Despite this, Eide and Johnston (1976) were able to provide reason- able predictions for the streamwise distribution of boundary layer integral thickness parameters in the upstream, unseparated sections of Moore's diffuser flows.Fortunately, there are two sets of experi- ments, Rothe and Johnston (1976) and Ibal and  Joubert (1993), on straight diffusers at high enough inlet aspect ratios so that end-wall secondary flow effects are of minor importance.
Rothe and Johnston (1975, 1976), used the fully- developed channel flow of Johnston et al. as a diffuser inlet condition.The diffuser side-wall length, L, and inlet throat width, W1, were fixed, but the side-walls were hinged at the inlet plane so that W2, the outlet width could be changed.Consequently, as the side-wall opening angle, 20, was changed the diffuser area ratio, AR--= W2/WI, could be varied.Studies were carried out with water flow which facilitated visualization using dye injection methods.Observation of the motion of the injected dye tracers permitted evaluation of the onset of the various stalled flow regimes (unseparated or no appreciable stall, corner stall, 2D stall, and full stall) as the magnitude of the as area ratio and rotation were increased in small steps.The results are shown in Fig. 11 where it is seen that the various regimes of stall set in at increasingly smaller area ratios as rotation rate increases.
Boundary layer separation at the upstream end of a stalled region, always occurred on the suction wall, the wall with the stabilized, lower turbulence boundary layer.Stall is first seen in the corners formed at the intersections of end-walls and the suction (stabilized) side-wall.When ROQ (defined below) increases at fixed AR (or 20) pockets of "corner stall" grow and then spread obliquely until the whole suction side-wall separates near the exit plane to form a "two-dimensional stall" zone.As ROQ continues to increase, the 2D separation line at the front of this zone moves upstream until it is very near the inlet plane.At this condition, we say the diffuser in a state of "full stall". .o AREA RATIO.AR FIGURE 12 Static pressure recovery coefficient, Cp, in ro- tating diffusers of Rothe and Johnston (1976).

EFFECTS OF SYSTEM ROTATION
In addition to the flow visualization to determine the stalled states, the static pressure distributions along the side-walls were also measured.From these data, the overall (outlet to inlet) pressure rise (recovery) was obtained.It is plotted versus AR in Fig. 12 in terms of the pressure rise coefficient from inlet to outlet: 2 Here, the inlet mean velocity was based on volume flow rate, Q, and inlet cross section area, Ul=Q/A].Also, the system Reynolds number and Rotation numbers were defined by ReQ U] WIly and ROQ f I/VII U] respectively.Pressure recovery attains a maximum (peak) value at a given area ratio for fixed Reynolds number and Rotation number, and the magnitude of Cp is reduced as Rotation number rises.As ROQ increases the peak is located at successively lower AR values; a trend that correlates with the results of the stall flow regime evaluation given in Fig. 11.
Ibal and Joubert (1993) investigated low speed (10m/s) air flows in diffusers that had thin, turbulent boundary layers at the inlet plane.They used the rotating boundary layer channel and general flow conditions of Watmuff et al. (see Section ii, above) to set up two cases: (i) a very lightly loaded, unstalled diffuser flow with opening angle, 20 3 , and (ii) a wider angle diffuser with 20 8.The wide angle diffuser had substantial 2D boundary separation on the suction (stabilized) surface when it rotated at 40 rpm, but it showed no separation when there was no rotation.This research study emphasized the detailed measure- ment of the growth of the side-wall boundary layers at several rotation rates in the adverse pressure gradients provided by the two diffuser configurations.In general, the observations and conclusions of Rothe and Johnston, and Watmuff et al. also apply here.
Observation #3 from Ibal and Joubert is worth quoting: "The separated region after two-dimen- sional separation was steady and relatively quiet." The same remark also applies to the results of Rothe and Johnston.It is of particular relevance for radial compressor rotor flows where steady separated regions called 'wakes' have been ob- served by various investigators, even in rotor passages of low aspect ratio.The reason for this behavior is that the free shear layer which separates the slowly recirculating fluid in the stalled region from the higher speed through flow** has the same sign of OU/Oy as the boundary layer on the suction ** In the literature on radial flow compressor rotor flows the separated region is call the 'wake' and the higher speed flow that passes over the wake region is called the 'jet', after the designations originated by Dr. R.C. Dean, Jr.
side, and thus it is stabilized by rotation, and at the operating Reynolds numbers it may never undergo free shear layer transition.This phenomenon will be discussed further below.
(iv) Back-Step Diffuser Flows with Separation Forced at the Step's Sharp Edge and Reattachment Downstream Back-step diffuser flows with separation forced at the sharp edge of the step, and reattachment on the downstream wall, behind the separated region were studied experimentally by Rothe and Johnston   (1979).The diffusers rotated about an axis perpen- dicular to their parallel end-walls, see (iv) in Fig. 6.In these cases, all the diffusers had the same area ratio (W2/W1 2), but each had a different aspect ratio (AS b/W1 2, 5, and 15).The experiments where carried out in our water flow channel at various Reynolds numbers from ReQ--3000-28,000, and for a range of Rotation number magnitudes; ]ROQI 0-0.15 (destabilizing), and [ROQ] 0-0.09 (stabilizing).
Stabilization is defined here, as above in terms of the sign of the velocity gradient across the free shear layer that divides the faster through flow from the slowly recirculating fluid in the separated region behind the step.When the free shear layer is destabilized, rotation causes the step side-wall to be a pressure (leading) side, and the boundary layers all along the step side-wall are also destabilized.The case is reversed when rotation is reversed to stabilize the free shear layer.In a faired wall dif- fuser, and in a radial flow rotor flow, only the stabilized situation exists as separation always starts in the suction (stable) side-wall boundary layer.
Visualization was conducted by injection of dye at the solid walls and by generation of streak lines using hydrogen bubbles generated from fine platinum wires.Analysis of hundreds of individual frames of movie film enabled us to determine the mean (average over a long time) location of reattachment, XR, at the downstream edge of a separation region.Fluctuations of XR, about the mean, were noticed even in stabilized cases because I-ooI Coriolis induced stabilization does not act to change the 2D, Kelvin-Helmholtz waves, it only damps three-dimensional, small-scale breakdown, the observed result of transition to turbulence in free shear layers.
The measured values of XR are plotted versus Rotation number in Fig. 13.With destabilizing rotation, turbulence in the free shear layer increases and entrainment of fluid from the recirculating separated region into the through flow is enhanced.As a result, the flow reattaches closer to the step, i.e., Xl decreases as [ROQI increases; a result that appears to be independent of diffuser aspect ratio (left hand side of Fig. 13).
On the other hand, the situation is considerably more complex for stabilized free-shear layers as a somewhat different pattern of events is seen for each aspect ratio (right hand side of Fig. 13).In this case, secondary flows near the end-walls are driven into the separated region, an effect opposite from the turbulent entrainment of fluid out of this region.Thus, secondary flows tend to increase, X, as more fluid is introduced into the separated region causing its volume to increase.This effect must be particularly strong at low aspect ratio where end-wall flows are more important, and it seems to justify the results in Fig. 13 where XR grows rather long for the case of AS--2.
With stabilization, only the AS 15 diffuser is essentially free of end-wall secondary flow effects.In this diffuser, the mixing layer remained laminar or was transitional at the lower Reynolds numbers and entrainment in these cases was .probablycontrolled by growth of the vorticies that are the result of the 2D Kelvin-Helmholtz instabilities.Consequently, at low Reynolds numbers, reattach- ment location did not change much from its location with zero rotation.One set of data, at ReQ 10,000 clearly shows an increase of XR as stabilizing ROQ increases, as though turbulence in the free shear layer were being damped.The range of Rotation numbers was too limited at the higher Reynolds numbers to draw any strong conclusions.
Nilsen and Anderson (1990) attempted to compute the higher Reynolds number flows for a step diffuser of infinite aspect ratio and AR 2, like one used in our experiments.They used the RANS equations with an algebraic, second order Reynolds stress closure model and obtained fair agreement with the experimental results as shown in Fig. 14, at the higher Reynolds numbers.Nilsen and Anderson   (1990).Data (open symbols) of Rothe and Johnston (1979), AS= 15, high Re numbers.*, Computed using algebraic sec- ond-moment Reynolds stress model, AS--c.
However, their results indicated a leveling (saturation) effect at high values of destabilization Rotation number.This effect was not seen in the Rothe and Johnston data, but a similar effect was seen in the wall friction data for the pressure (destabilized) side of the rotating channels (upper branch in Fig. 8).In both cases, this saturation of the destabilizing effect of rotation was attributed to a rather complex interation of the quasi stationary roll cell structures with the destabilizing influence of rotation on the production terms in the equations for the evolution of the turbulence Reynolds stresses (derived from the Navier-Stokes equations).A detailed examination of their papers is suggested for modelers interested in developing practical methods of computation.
3. CONCLUDING REMARKS An attempt was made to conduct a coherent discussion of two of the effects of system rotation (secondary flow and stabilization) on flows in geometrically simple cases, rectangular channels and diffusers.The effects rotation in regard to secondary flows in end-wall boundary layers were already fairly well understood 25-30 years ago.
However, new, basic data on the stabilizing/ destabilizing effects of Coriolis force on boundary and free shear layers has been obtained since 1970, and thus primary emphasis was placed on discus- sion of the latter effects in this paper.
Although the review has concentrated on experi- ments in simple basic flows, the discussion indi- cates where and how these results are related to practical flows in pump and compressor impellers and turbine rotors.Predictions of complex flows in rotors is a major objective of engineers and designers, and some have attempted to empirically model the stabilizing/destabilizing effects for inclu- sion in codes that numerically solve the Reynolds averaged Navier-Stokes equations.The state of the art in practical turbulence modeling for turbo- machines, or for that matter even simple rotating flows such as straight channels and diffusers with wall boundary layers is still rather primitive, and thus this topic is better left to a substantial review at a later time, hopefully sooner than 25 years from now.

E EN NE ER RG GY Y M MA AT TE ER RI IA AL LS S Materials Science & Engineering for Energy Systems
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FIGURE 5
FIGURE5 Coriolis force on a fluid particle near a suction surface where w is effective rate of rotation for stability effects.

Figure 6
Figure6illustrates the general configurations of these four situations.

FIGURE 10
FIGURE 10 Variation of modified log law constant, from Watmuff et al. (1985) over wide range of conditions.
FIGURE 13 Reattachment distance downstream of the step in step diffusers of Rothe and Johnston (1979) versus Rota- tion number.Reattachment distance, XR, is normalized on step height, b.

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