Local normal-fluid helium II flow due to mutual friction interaction with the superfluid

Transcription

1 PHYSICAL REVIEW B VOLUME 62, NUMBER 5 1 AUGUST 2000-I Local normal-fluid helium II flow due to mutual friction interaction with the superfluid Olusola C. Idowu, Ashley Willis, Carlo F. Barenghi, and David C. Samuels Department of Mathematics, University of Newcastle, Newcastle upon Tyne NE1 7RU, United Kingdom Received 19 November 1999; revised manuscript received 14 April 2000 Mutual friction, the scattering of rotons and phonons by quantized vortex lines, acts locally on the normal fluid component of helium II and is nonzero only in a volume very close to the quantized vortex line. We show that this localized mutual friction force forms a jetlike flow structure in the normal fluid, with a vortex dipole structure. The peak velocity of this normal fluid jet is a significant fraction of the velocity of the quantized vortex line and the width of the jet is on the order of 0.1 mm. The normal fluid jet velocity is highest at T 1.9 K and decreases to zero as the temperature goes to zero and also as the temperature approaches the lambda transition temperature. We report the circulation, energy per unit length, and the viscous dissipation of kinetic energy of this normal fluid flow structure as a function of temperature. I. INTRODUCTION The flows of the superfluid and normal fluid components of helium II are coupled together by pressure and temperature gradients, and by mutual friction. In this paper we concentrate on the least understood of these couplings, the mutual friction coupling. Mutual friction occurs when the excitations phonons and rotons that make up the normal fluid component are scattered by the presence of quantized vortex filaments in the superfluid. 1,2 The phonons and rotons are scattered partially by the superfluid vortex filament core, at a length scale of 1 Å, and partially by the superfluid velocity gradients surrounding the vortex filament, 2 on a length scale of a few hundred Å. In both cases the length scale over which the mutual friction force acts on the normal fluid is quite small and we can consider the mutual friction force on the normal fluid to be approximately a delta function force, nonzero only along the one-dimensional superfluid vortex lines embedded in the three-dimensional normal fluid flow. While the effect of mutual friction on the flow of the superfluid and the motion of the superfluid vortex lines is well understood, 1,3 the effect of this force on the normal fluid flow is not well understood. In the 1950 s, Hall and Vinen 4,5 showed that the mutual friction from a moving superfluid vortex filament would locally drag the normal fluid. The arguements in these papers are insightful and elegant, but with modern computer power we can improve our understanding of this problem through a direct numerical simulation. In this paper we use the full Navier-Stokes equation for the normal fluid, including a localized mutual friction force, and we calculate the spatial structure of the resulting normal fluid flow. We also examine the detailed characteristics of this flow: the circulation, kinetic energy, and viscous energy dissipation. II. BASIC EQUATIONS AND NUMERICAL METHODS Our simulation calculates the coupled motion of a twodimensional 2D normal fluid flow with one or more superfluid vortex points cross sections through the superfluid vortex lines. This is a fully coupled, self-consistent two-fluid calculation, solving for both the normal fluid velocity v n (x,t) and the motion of the superfluid vortex filaments. We consider only isothermal situations with no external pressure gradients, so that the only interaction between the two fluids is through the mutual friction force. Restricting our flow to the 2D case still captures the most essential physics of the problem in our opinion and allows us to use a reasonably high grid resolution. Work on the corresponding threedimensional problem is underway. The velocity of the superfluid vortex line V L is 6 V L h 1 V s V I h 2 tˆ v n V s V I h 3 v n, where h 1 s D 0 D 0 2 D 2, h 2 s D D 0 2 D 2, h 3 D2 D 0 D t D 0 2 D 2, 4 D 0 s D t. 5 D and D t are mutual friction coefficients, s is the superfluid density, is the quantum of circulation, V I is the velocity induced by the presence of any superfluid vortex filaments, and V S is any externally applied superfluid velocity field. For simplicity, in this paper we will take V S to be a uniform constant and we will consider only single vortex filaments vortex points in 2D, sov I 0. The unit vector tˆ is the direction vector of each superfluid vortex either ẑ or ẑ in the 2D case. Readers familiar with the literature on superfluid vortices 7 will immediately notice that this equation of motion is not the same as the one that is used in calculations where the normal fluid flow is held fixed. Since we include the local motion of the normal fluid (v n ) in response to the superfluid vortex, we must use a mutual friction force based on this local velocity, not the mutual friction force based on the averaged velocity. This change in the mutual friction force leads to a change in the form of the equations of motion for the superfluid vortex. A detailed discussion /2000/62 5 / /$15.00 PRB The American Physical Society

2 3410 IDOWU, WILLIS, BARENGHI, AND SAMUELS PRB 62 and derivation of this equation of motion is given in Ref. 6. In place of the more familiar parameters and, this formulation of mutual friction involves the parameters D and D t, which are the first and second coefficients of mutual friction. 1,8 For the relationship between D and D t and the coefficients and see Refs. 1, 3, 4, and 6. We solve for the normal fluid flow v n in two dimensions by a stream function method. We define the stream function n by v n n y, n x,0. Using this definition, the stream function obeys a Poisson equation 2 n n,z 0, where n,z is the z component of vorticity in the normal fluid. We assume periodic boundary conditions for n and n,z. The equation for the vorticity vector n is 9 n n v t n n n v n 2 n F mf, 8 where n is the normal fluid density and is the normal fluid viscosity. In the 2D flow only the ẑ component of this equation is nonzero and the vortex stretching term disappears. The mutual friction force per unit area F mf exerted on the normal fluid by the superfluid vortex points is 9,10 F mf D A tˆ tˆ vn V S V I D t tˆ vn V A S V I, 9 where A is the area of the normal fluid over which the mutual friction force is distributed. The sum is taken over all superfluid vortex points within the area A. The force F mf is a function of position and is nonzero only in areas of size A taken to be the computational grid spacing which contain superfluid vortex points. The values of the temperature dependent parameters in Eqs. 1, 8, and 9 were taken from Ref. 8. The vorticity obtained at a time t from Eq. 8 is used to calculate the stream function of the normal fluid at the time t t by inverting Eq. 7. Concurrently, we solve for the motion of the superfluid vortex points, Eq. 1, with the Euler method. With two simultaneous calculations for the normal fluid and the superfluid, we must choose a time step to satisfy both calculations. One limit on the time step is that we must keep it below the limit set by the normal fluid viscosity t visc ( x) 2 n /, where x is the grid spacing. This is a necessary stability criterion for finite difference schemes. 12 We want the superfluid vortex points to be moving smoothly through the normal fluid computational grid, so that sets another limit t sf x/v L, where V L is the magnitude of the velocity of motion of the superfluid vortex point. We set the time step of the calculation to be 1/10 of the minimum of these two values. Typically, the normal fluid calculation was 6 7 done on a grid with a physical size of 1 mm 2. Tests of the code were run using both larger and smaller grids. Ideally, the mutual friction force on the normal fluid from a single superfluid vortex point would be a delta function forcing at the position of the superfluid vortex. To model this localized forcing, we distributed the mutual friction force over the four corner grid points of any grid space containing a superfluid vortex point. This sets the area in Eq. 9 to be A ( x) 2. As the superfluid vortex point moves through one grid space and into a neighboring grid space the mutual friction force on the normal fluid also moves. To make this transition of the mutual friction force between the grid spaces smooth we weighted the force distribution on the four corner points. We chose a simple linear weighting w ij (1 x ij )(1 y ij ), where x ij and y ij are the x and y distance, respectively, of the superfluid vortex point from each of the four corner grid points. The sum of these four weightings is one. With this method the mutual friction forces on the normal fluid grid always change continuously as the superfluid vortex point moves across the grid. We tested this model by changing the grid spacing and we saw no strong effect of the grid spacing on the v n generated by the mutual friction forcing. These tests are discussed in Sec. IV. III. INDUCED FLOW IN THE NORMAL FLUID The results of this calculation of v n are shown in Fig. 1. A localized jet structure forms in the normal fluid in response to the mutual friction force. A low velocity recirculation flow probably due to the periodic boundary conditions can be seen around the edges of the computational box Fig. 1 A. The center of the normal fluid jet corresponds to the position of the superfluid vortex line, and moves with it, but the direction of the jet does not necessarily coincide with the direction of the superfluid vortex motion. We will discuss this point later in this paper. The spatial structure of this jet can be more clearly seen in Fig. 1 B, a plot of the velocity magnitude of the normal fluid. The jet velocity is sharply peaked toward the center and the shape of the jet is slightly elongated in the direction of the jet velocity. The vorticity of the normal fluid flow Fig. 1 C has a clear dipole structure. This can also be seen in the streamlines of the normal fluid flow Fig. 1 D. The characteristic shape of the normal fluid flow, illustrated in Fig. 1, was unchanged throughout the temperature range explored, 1.3 K T K slightly below 8 the lambda transition temperature T ). The dependence of the magnitude of the peak velocity of the normal fluid jet on temperature and the superfluid vortex velocity V L is given in Fig. 2. These simulations were started with the normal fluid at rest and the superfluid vortex moving according to Eq. 1 under an external superfluid velocity V S. The normal fluid jet typically formed on time scales of approximately 0.1 s, and the value for the peak velocity was taken at 1 s, well after the initial transients had died down. For temperatures in the range 1.6 K T 2.1 K the peak jet velocity reaches a significant fraction of the driving superfluid velocity but for T 1.6 K and T 2.1 K the peak normal fluid velocity decreases to low values. Even though the peak normal fluid velocity is always smaller than the velocity of the superfluid vortex point, the position of the peak nor-

3 PRB 62 LOCAL NORMAL-FLUID HELIUM II FLOW DUE TO FIG. 1. Velocity field of the normal fluid due to the mutual friction forcing of a single superfluid vortex. The calculation was made on a grid. The size of the computational box is 1 mm. The temperature dependant parameters were taken at T 1.9 K and a uniform superfluid velocity of V S 10 2 cm/s moves the superfluid vortex. At the time when these data were taken the superfluid vortex was located at the grid position 77,65 and was moving in the xˆ direction. A Velocity arrow plot of v n. B Magnitude of v n normalized by the vortex line velocity V L. C Normal fluid vorticity n. The normal fluid vorticity has the form of a vortex dipole. D Streamlines of the normal fluid flow. mal fluid velocity Fig. 1 always remains centered on the position of the superfluid vortex. This correspondence points out that the normal fluid jet is continually being generated by the mutual friction force with the superfluid vortex and is rapidly dissipated by viscosity as the superfluid vortex moves ahead of the induced normal fluid flow. The temperature and V S dependence of the peak velocity in the normal fluid can be understood by a simple model comparing the magnitudes of the viscous and the mutual friction terms in the Navier Stokes equation for v n.ifwe describe the normal fluid jet by a velocity scale V peak, the peak velocity of the jet, and a length scale L describing the jet size, then the magnitude of the viscous term integrated over an area of size A L 2 can be approximated as 2 v n da L 2 V peak L 2 V peak. 10 The integration over the area L 2 is needed to convert the delta function form of the mutual friction force density to a finite magnitude force. The magnitude of the mutual friction force is F mf da D 2 D 2 t L 2 V peak V L L 2 D V peak V L. 11 If the mutual friction force and the normal fluid viscosity are the dominant physical processes in this normal fluid jet, then the peak velocity V peak of the jet can be calculated by setting 2 v n da C F mf da, where C is a constant of order unity. Doing this, and solving for V peak we have V peak V L 1 CD The values of the constant C were set by a least square fit of the simulation results with the assumption that C is independent of temperature, though we do find that our fit values of C are slightly dependent on V S Fig. 2. This simple model

4 3412 IDOWU, WILLIS, BARENGHI, AND SAMUELS PRB 62 FIG. 2. Peak velocity of the normal fluid jet vs temperature, for different superfluid velocities V S. The results of Eq. 12 are plotted as the solid lines with the value of the parameter C set by a least squares fit. works quite well for the lower values of V S but does not work well for the highest velocity data shown in Fig. 2. It would be reasonable that at these higher velocities the nonlinear term in the Navier Stokes equation may become nonnegligable and this simple model would no longer hold. An equation of the same form as Eq. 12 can also be derived from the results of Hall and Vinen. 4 From Eqs. 2, 4, and 5 of that paper one can derive V R V L 1 MD 4 1, 13 where V R is defined by Hall and Vinen as the average drift velocity of the rotons that collide with vortex lines. The parameter M is defined as FIG. 3. The length L and width L of the normal fluid jet as a function of temperature and superfluid velocity V S. The length scales are defined as the distance over which the velocity falls to 25% of the peak velocity. M Real 1 ln l /2 i /4 14 where l is the roton roton mean free path which is of order 1nm 1,13 in the temperature range 1.3 K T T. The effective viscous penetration depth in the zero frequency limit 5,11 is 2 / n v n V L. Note that M is negative since l in the temperature range that we are considering. If we identify the roton drift velocity V R with our local peak velocity of the normal fluid V peak then the correspondance between Eqs. 12 and 13 is very strong. Equation 13 does give values for V R /V L that are slightly closer to unity than are our simulation results for V peak /V L. For such a simple model Eq. 12 explains the temperature dependence of the peak jet velocity remarkably well for values of V S 1 cm/s. To gain a perspective on the magnitude of V S, if we consider the superfluid velocity to be that induced by a neighboring superfluid vortex filament, then a velocity of 0.1 cm/s would be induced by a filament only a distance of 0.02 mm away. So the magnitudes of V S used in Fig. 2 are not unreasonably small. The simple model given above predicts the temperature dependence of the peak velocity of the jet well, but says nothing about the length scale of the jet because that scale L cancelled out in Eq. 10. Since the normal fluid jet is elongated we must define two length scales, a small length scale transverse to the direction of the jet L and a longer length scale parallel to the jet direction L. A practical, but arbitrary, definition of these length scales is to take the width and length of the jet to be where the velocity magnitude falls to 25% of the peak velocity. To simplify the measurements in the parallel and perpendicular directions the applied velocity V S was set at an angle from the xˆ axis, with the angle chosen so that the direction of the resulting normal fluid jet was aligned with the xˆ axis of the grid. These values of the normal fluid jet length scales are plotted in Fig. 3. If we allowed the superfluid vortex point to cross through the periodic box many times, always crossing over the same sections of the grid, we observed that a low level normal fluid flow would build up after repeated crossings of the filament through the grid. This behavior is apparently a numerical artifact of the periodic grid, and although it was only a low level velocity it was sufficient to disturb the measurements of the length scales since our measurement criteria were sensitive to changes in low velocity levels. With these in mind, we only allowed the vortex filaments to move through the computational box once, and then took the length data. This did mean that we could not collect these data for the higher values of V S, since those filaments would move through the computational box once in a time faster than that needed to establish the final steady flow in the normal fluid. Neither the width nor the length of the normal fluid flow structure shows a strong temperature dependence except in the temperature range very close to the lambda point. The width of the normal fluid jet is L 0.1 mm and the length is L 0.4 mm. We do not currently have any predictive theory for these length scales. But we can point out that neither of these length scales can be explained by a simple definition of a viscous length scale, /(V peak n ) since this viscous length scale is strongly dependent on the velocity V peak while our measured values are not, and it is much more strongly temperature dependent than the values in Fig. 3. The normal fluid flow develops in response to the mutual friction force. This force Eq. 9 has components both parallel and perpindicular to the motion of the superfluid vortex filament. So in principle, the normal fluid velocity v n at the position of the vortex filament and the vortex line velocity V L may point in different directions with an angle separating them. From the form of Eq. 9 one would expect that this angle would be given by

5 PRB 62 LOCAL NORMAL-FLUID HELIUM II FLOW DUE TO FIG. 4. The angle between the peak normal fluid velocity and the superfluid line velocity for a positively oriented superfluid vortex. tan 1 D t /D. 15 We have measured this angle in our simulations Fig. 4 and these values are in good agreement with Eq. 15. Inthe medium temperature range, where the peak normal fluid velocity is highest, the angle is very small, and the normal fluid jet points almost in the same direction as the motion of the superfluid filament. At temperatures approaching T the angle becomes large, and the velocity generated in the normal fluid points almost at right angles to the motion of the superfluid vortex and thus to the motion of the normal fluid flow structure itself. This large angle is due to the rapidly increasing value of the ratio of the mutual friction parameters D t /D as T T. IV. TESTS FIG. 5. The circulation around one side of the normal fluid vortex dipole, for a range of superfluid velocities. For comparison, the dashed line marks the quantum of circulation in the superfluid. We carried out a series of tests to see if our measurements of the normal fluid flow structure were sensitive to the numerical parameters such as grid spacing and time step. We tested the simulation results for numerical artifacts by changing the grid size and the length scales of the computational box. Through trial and error we found that a computational box size of 1 mm was a good compromise choice. This choice lowered the effect of the periodic boundary conditions by keeping the box size large in comparison to the size of the normal fluid flow structure and it also allowed sufficient grid resolution over the flow structure using a practical grid of points. Tests with the box size doubled to 2 mm and the grid enlarged to thus keeping the same grid spacing x) raised V peak by 9%, and L and L by 9%, of the results in Fig. 2 and 3. This shows that there are no significant effects of the computational box size or the periodic boundary conditions on the velocity magnitudes or on the length scales of the jet. Tests with the higher resolution mesh, but keeping the same box size of 1 mm thus lowering the grid spacing x), raised the measured peak velocity slightly by 18% of the value reported in Fig. 2 and lowered both the length scales by approximately 18%. This tests the effect of the area A x 2 in Eq. 9 on the normal fluid flow structure. Thus the true peak velocity that measured with an infinitely fine mesh in the normal fluid jet would be slightly higher than that calculated in our simulations. Since the time needed for each simulation typically increases as N 4 for an N N grid, it was not practical for us to take data at all temperatures for grids finer than We tested the numerical effects of the time step on the results by lowering the allowed time steps described in Sec. II by a factor of 10. There were no significant changes in the results obtained. V. NORMAL FLUID CIRCULATION, KINETIC ENERGY, AND VISCOUS DISSIPATION The normal fluid velocity structure has a dipole vorticity form, so the circulation defined as n,z da) is zero when the integral is taken over the entire computational area. But we can measure a circulation value by taking the integral over only half of the area. Facing in the direction of motion of the flow structure, the vorticity to the left is always positive and the vorticity to the right is always negative Fig. 1 C. We can define a positive circulation for this structure by integrating only over the area that contains positive vorticity da, 16 where the subscript means that we keep only the positive values of vorticity and set all negative values to zero in the integral. We can similarly define a negative circulation by integrating only over the area with negative vorticity. We always find that to very good precision, confirming the dipole nature of the normal fluid vorticity. The measured values of this circulation are presented in Fig. 5. In the range of velocities V S considered, the circulation scale of the normal fluid is roughly of the same order as the superfluid quantum of circulation. But since our simulation results show that the circulation scale is dependent on the superfluid velocity V S, then could be very different from outside this velocity range. Our calculations show that the normal fluid circulation is approximately proportional to V S. The kinetic energy per unit length with length extending in the ẑ direction of the normal fluid flow is defined simply as E n (1/2) n v n 2 da Fig. 6. For comparison, we have

6 3414 IDOWU, WILLIS, BARENGHI, AND SAMUELS PRB 62 FIG. 6. The kinetic energy per unit length of the normal fluid flow structure. For comparison, the dashed line is the kinetic energy per unit length of the superfluid quantized vortex filament. also plotted the energy per unit length of the superfluid vortex line, defined as E s ( s 2 /4 )ln(b/a), where b is the size of our computational box and a is the superfluid vortex core length scale. 1 Our calculated values of E n are approximately proportional to V 2 S. Even though the normal fluid circulation scale for these flow structures is of the same order as the superfluid quantum of circulation, the kinetic energy of these flows is orders of magnitude lower than that of the superfluid vortex. This is due to the vortex dipole nature of the normal fluid flow. The opposite vorticities shield each other, so that the resulting velocity field is much weaker than that around a vortex line and this greatly reduces the normal fluid energy per unit length. The energy to form the normal fluid flow comes from the superfluid flow, through mutual friction, but we must point out here that this energy does not come from the energy per unit length of the individual superfluid vortex line. That quantity is set by the quantum of circulation and s and cannot change. In these simulations, the normal fluid energy comes from the imposed superfluid flow that gives the flow V S that moves the superfluid vortex relative to the normal fluid. Since this V S is held constant, it acts as an infinite source of energy. If we had a pair of superfluid vortices, with each vortex moving due to the velocity field of the other, then the energy source for the formation of the normal fluid flow structures would be the energy of the pair of superfluid vortices. The viscous dissipation of kinetic energy per unit length is de dt 2 n e 2 da 17 where e ij is the rate of strain tensor for the normal fluid. This quantity is plotted in Fig. 7. Again, these values are approximately proportional to V S 2. The ratio of the viscous dissipation to the kinetic energy of the flow defines a time scale for the decay of the flow if all energy sources are removed. For the normal fluid flows presented in Figs. 6 and 7, this time scale is of order 0.1 s and is almost independent of the superfluid velocity V S since both quantities in the ratio are approximately proportional to V S 2 ). The normal fluid flows FIG. 7. The kinetic energy dissipation per unit length of the normal fluid flow structure. described in this paper are very dissipative and they maintain a steady flow only because they are constantly renewed by the energy gained from the superfluid through mutual friction. This point is reinforced by the observation in the simulations that the normal fluid flow structure moves along with the superfluid vortex, at a velocity V L, even when the peak normal fluid velocity V peak is always lower than V L and may even be pointing in a different direction Fig. 4. The normal fluid flow structure is being constantly generated as the superfluid vortex moves into stationary normal fluid, and the normal fluid flow left behind the superfluid vortex decays away rapidly leaving little, if any, appreciable velocity in the wake of the superfluid vortex Fig. 1. VI. CONCLUSIONS From the results of these simulations we conclude that each superfluid vortex line moving relative to the normal fluid will locally drag the normal fluid along with it. The maximum dragging is approximately 55 60% of the velocity of the superfluid vortex line and occurs in the temperature range 1.6 K T 2.1 K. By balancing the magnitudes of the viscous term and the mutual friction term in the normal fluid velocity equation we have derived an equation which explains the temperature dependence of the induced normal fluid velocity quite well. At temperatures below about 1.6 K the decreasing value of the mutual friction parameter D causes the ratio V peak /V L to drop. At temperatures between about 2.1 K and T the parameter D again decreases, dropping the induced normal fluid velocity to zero. The temperature dependence of the normal fluid viscosity also helps to decrease the induced normal fluid velocity at both temperature extremes. The width of the normal fluid jet is approximately 0.1 mm and the length is approximately 0.4 mm and these values have only a slight temperature dependence. While these length scales are small, helium II hydrodynamics experiments are often carried out in small geometries and it is conceivable that some method of detecting the affect of normal fluid flows on this scale could be devised. For example, helium II Couette flow experiments 14 have been conducted with a gap of only 0.5 mm width between the concentric cylinders. This is only a factor the of 2 5 times larger than

7 PRB 62 LOCAL NORMAL-FLUID HELIUM II FLOW DUE TO the length scales that we observe in the normal fluid flow Fig. 3. It is then reasonable to expect that the normal fluid flow structures described in this paper could have a measurable effect in such experiments. We have no theoretical explanation for these length scales, but we can point out that neither the weak temperature dependence nor the weak velocity dependence are consistent with a simple definition of a viscous length scale. It is interesting that the normal fluid flow in Fig. 1 can be interpreted as a vorticity dipole. Thus a superfluid vorticity monopole the vortex filament induces a vorticity dipole in the normal fluid. The two fluids form a strongly coupled triple vortex system, consisting of a central superfluid vortex filament flanked by two concentrations of normal fluid vorticity, of opposite circulation to each other. In the medium temperature range, where the response of the normal fluid to the mutual friction forcing is largest, the parameter D is much larger than D t, so the D term is the dominate term in the mutual friction on the normal fluid. 6 Since this term is quadratic in the superfluid vortex direction vector tˆ then the same flows will be induced in the normal fluid by both positively and negatively oriented superfluid vortices. Facing in the direction of motion of the superfluid vortex filament, any superfluid vortex filament will induce a positive normal fluid vorticity to the left side and a negative normal fluid vorticity to the right side see Fig. 1. The direction of the angle between the peak v n and the velocity of the vortex line V L Fig. 4 will depend on the sign of the superfluid vortex, but this angle is only significant at temperatures approaching the lambda transition. Finally, we should consider the consequences of these results for helium II turbulence. 15,16 Without the mutual friction forcing, the normal fluid obeys a Navier Stokes equation. In this case we would expect the turbulence in the normal fluid component of helium II to be completely classical. But the introduction of the mutual friction forcing from the superfluid may change the character of the normal fluid turbulence. The localized normal fluid flow structures jets formed by the mutual friction forcing have an unusual spatial form since they have small length scales in two dimensions perpendicular to the superfluid vortex lines but are indefinitely long in the direction parallel to the vortex lines. This could be considered as a forcing of the normal fluid turbulence, via mutual friction, over a large range of spatial scales. This type of forcing, which is quite different from the typical turbulent flow forcing usually confined to large length scales, may affect the characteristics of the normal fluid turbulence in unforeseen ways. One prediction that we can make is that the effect of the mutual friction on the normal fluid flow is likely to be strongest in a medium range of temperature, between 1.6 and 2.1 K, and weaker at both higher and lower temperatures. ACKNOWLEDGMENTS O.C.I. is supported by a grant from the University of Newcastle. A.W. was suported by a Undergraduate Research Bursary from the Nuffield Foundation. 1 C.F. Barenghi, R.J. Donnelly, and W.F. Vinen, J. Low Temp. Phys. 52, D.C. Samuels and R.J. Donnelly, Phys. Rev. Lett. 65, R.J. Donnelly, Quantized Vortices in Helium II Cambridge University Press, Cambridge, H.E. Hall and W.F. Vinen, Proc. R. Soc. London, Ser. A 238, W.F. Vinen, Proc. R. Soc. London, Ser. A 242, O.C. Idowu, D. Kivotides, C.F. Barenghi, and D.C. Samuels, J. Low Temp. Phys. to be published August K.W. Schwarz, Phys. Rev. B 31, R.J. Donnelly and C.F. Barenghi, J. Phys. Chem. Ref. Data 27, C.F. Barenghi and D.C. Samuels, Phys. Rev. B 60, R.G.K.M. Aarts and A.T.A.M. dewaele, Phys. Rev. B 50, C.E. Swanson, W.T. Wagner, R.J. Donnelly, and C.F. Barenghi, J. Low Temp. Phys. 66, R. Peyret and T.D. Taylor, Computational Methods for Fluid Flow Springer, New York, R.J. Miller, I.H. Lynall, and J.B. Mehl, Phys. Rev. B 17, C.J. Swanson and R.J. Donnelly, Phys. Rev. Lett. 67, ; C.F. Barenghi, Phys. Rev. B 45, R.J. Donnelly, High Reynolds Number Flows Using Liquid and Gaseous Helium Springer, New York, R.J. Donnelly and K.R. Sreenivasan, Flow at Ultra-High Reynolds and Rayleigh Number Springer, New York, 1998.