Flow around a rotating, semi-infinite cylinder in an axial stream

Abstract

This paper concerns steady, high-Reynolds-number flow around a semi-infinite, rotating cylinder placed in an axial stream and uses boundary-layer type of equations which apply even when the boundary-layer thickness is comparable to the cylinder radius, as indeed it is at large enough downstream distances. At large rotation rates, it is found that a wall jet appears over a certain range of downstream locations. This jet strengthens with increasing rotation, but first strengthens then weakens as downstream distance increases, eventually disappearing, so the flow recovers a profile qualitatively similar to a classical boundary layer. The asymptotic solution at large streamwise distances is obtained as an expansion in inverse powers of the logarithm of the distance. It is found that the asymptotic radial and axial velocity components are the same as for a non-rotating cylinder, to all orders in this expansion.

1. Introduction

When a semi-infinite rotating cylindrical body is placed in a high-Reynolds-number axial flow (figure 1), an axisymmetric boundary layer develops along the cylinder. Initially thin, this layer becomes of thickness comparable with, then larger than the cylinder radius at sufficiently large axial distances. Our original motivation for studying this flow was to undertake a stability analysis. However, it soon became clear that there are very few existing studies of the underlying flow, despite its interesting features, e.g. the appearance of an axial wall jet beyond a certain threshold value of the rotation rate. The presence of curvature and rotation means that the classical Prandtl equations need to be generalized to allow for these effects. In particular, rotation leads to a centrifugal term which couples all three velocity components. This results in significant qualitative changes in the flow structure, e.g. the wall jet, compared with Blasius flow on a flat plate.

Figure 1.

Schematic diagram of the problem.

The non-rotating version of this problem was studied analytically by Seban & Bond [1] using a series solution in powers of z^1/2, where z is the axial coordinate, non-dimensionalized using the cylinder radius. This series solution was limited to order 3, and thus only applicable close to the inlet. Kelly [2] showed that the series solution for the displacement thickness provided by Seban & Bond [1] was erroneous, and obtained the correct result. Glauert & Lighthill [3] extended this work to obtain a solution at all z using the Pohlhausen approximation. At large z, Glauert & Lighthill [3] also showed that the velocity profile had an asymptotic expansion in inverse powers of $\log (z)$. Jaffe & Okamura [4] were the first to solve the boundary-layer equations for this problem numerically, thus covering the entire range of z, from small to large values. Boundary-layer velocity profiles have also been numerically determined by Tutty et al. [5] and Vinod & Govindarajan [6] in the context of stability analysis.

Petrov [7] appears to be the first to have studied the rotating case. Axial velocity profiles were obtained in the limit of small z and show the existence of a wall jet for sufficiently strong rotation, though this interesting feature was not explained in the paper. Motivated as we were by stability analysis of the flow, Kao & Chow [8] and Herrada et al. [9] solved the present problem numerically. However, both papers limit themselves to a range of rotation rates insufficiently large to produce a wall jet. Furthermore, the centrifugal term is missing in the boundary-layer equations of Kao & Chow [8], and so they are incapable of yielding a wall jet even at large rotation rates.

In §2, we define the two non-dimensional control parameters of the problem, Re and S, the Reynolds number and non-dimensional rotation rate. The boundary-layer equations, valid for large Re, and allowing for boundary-layer thickness to be comparable with the cylinder radius are given. These equations generalize the Prandtl equations and apply for arbitrary (not necessarily small) ratios of boundary-layer thickness to cylinder radius. Suitable rescaling of the variables renders the problem independent of Re, leaving only S as control parameter. Section 3 describes the numerical scheme and its verification, while §4 gives results and discussion, in particular focusing on the wall jet. Finally, §5 gives asymptotic analyses of the limits of large Z=z/Re and large S. The boundary layer on the nose is discussed in the appendices. It is found that the precise shape of the nose is unimportant: the input to the boundary-layer equations of §2 being the Blasius flat-plate flow (generalized to include the azimuthal component due to rotation), independent of the nose shape.

2. Boundary-layer equations

A semi-infinite cylinder of radius a rotates about its axis with angular velocity Ω and is placed in an axial stream of incompressible fluid of velocity $U_{\mathrm{\infty }}$ and viscosity ν (figure 1). Assuming large Reynolds number, an initially thin boundary layer develops along the cylinder. If the cylinder were sharply truncated at the nose, flow separation would occur as is usually the case at a salient edge [10]. To avoid this scenario, we assume that there is a smooth nose, as shown in figure 1. Another way to avoid flow separation would be to consider a thin hollow cylinder. The boundary-layer equations formulated in this section hold good for both these cases. A detailed analysis of the nose region is given in appendix B.

The natural length and velocity scales are a and $U_{\mathrm{\infty }}$. These scales are used to non-dimensionalize the axisymmetric, steady Navier–Stokes equations in cylindrical coordinates, z,r,θ. There are two non-dimensional parameters, namely the Reynolds number

$$Re=\frac{U_{\mathrm{\infty }}a}{\nu },$$ 2.1

and the rotation rate

$$S=\frac{\mathrm{\Omega }a}{U_{\mathrm{\infty }}}.$$ 2.2

Assuming a large Reynolds number, the length scale for axial variation of the flow is much longer than that for radial variation. This separation of scales leads to the boundary-layer approximation. Thus,

$$\begin{array}{rl}& u_z\frac{\mathrm{\partial }{u}_z}{\mathrm{\partial }z}+u_r\frac{\mathrm{\partial }{u}_z}{\mathrm{\partial }r}=-\frac{\mathrm{\partial }p}{\mathrm{\partial }z}+\frac{1}{Re}(\frac{{\mathrm{\partial }}^2{u}_z}{\mathrm{\partial }{r}^2}+\frac{1}{r}\frac{\mathrm{\partial }{u}_z}{\mathrm{\partial }r}),\end{array}$$ 2.3

$$\begin{array}{rl}& \frac{u_{\theta }^2}{r}=\frac{\mathrm{\partial }p}{\mathrm{\partial }r},\end{array}$$ 2.4

$$\begin{array}{rl}& u_z\frac{\mathrm{\partial }{u}_{\theta }}{\mathrm{\partial }z}+u_r\frac{\mathrm{\partial }{u}_{\theta }}{\mathrm{\partial }r}+\frac{u_{\theta }{u}_r}{r}=\frac{1}{Re}(\frac{{\mathrm{\partial }}^2{u}_{\theta }}{\mathrm{\partial }{r}^2}+\frac{1}{r}\frac{\mathrm{\partial }{u}_{\theta }}{\mathrm{\partial }r}-\frac{u_{\theta }}{r^2})\end{array}$$ 2.5

$$\begin{array}{rl}\text{and}& \frac{\mathrm{\partial }{u}_z}{\mathrm{\partial }z}+\frac{1}{r}\frac{\mathrm{\partial }r{u}_r}{\mathrm{\partial }r}=0\end{array}$$ 2.6

are obtained by dropping terms of higher order from the Navier–Stokes equations in the usual manner. Note that we have not assumed the boundary layer to be thin compared with the radius. The boundary conditions are

$$u_z=0,u_r=0,u_{\theta }=Sz>0,\hspace{0.17em}r=1$$ 2.7

and

$$u_z\to 1,u_{\theta }\to 0,p\to 0z>0,\hspace{0.17em}r\to \mathrm{\infty }.$$ 2.8

The above equations contain the azimuthal component, u_θ, of the velocity. This is due to rotation of the cylinder, which induces the centrifugal term on the left-hand side of equation (2.4), leading to a significant radial pressure gradient. Such an effect is not present in classical boundary-layer theory, which predicts near constancy of the pressure across the layer. Compared to the Prandtl equations of a classical boundary layer, equations (2.3)–(2.6) allow for the additional effects of both rotation and curvature. Near the nose the boundary layer is thin compared with the cylinder radius and curvature effects are negligible. But at large z, boundary-thickening eventually makes the thickness comparable to, then larger compared with the radius, and the full set of equations is required.

The above problem needs to be completed by inlet conditions. At distances from the nose of O(a), the boundary layer is thin compared with the radius and is described by the axisymmetric Prandtl equations given in appendix B. On the cylinder (after leaving the nose), these equations become the flat-plate Prandtl equations, and as z increases, we expect the flow to forget the precise initial conditions and to approach the Blasius solution, independent of the nose shape (here, we implicitly suppose the nose length to be of the same order as its diameter). There are, in fact, two asymptotic regions, z=O(1), where the equations of appendix B apply, and z=O(Re), where the boundary-layer thickness is comparable to the cylinder radius and equations (2.3)–(2.8) hold. Matching between these regions requires that the inlet condition be the Blasius solution (generalized to include the azimuthal component due to rotation). The same is true for the case of the hollow cylinder. In either case, equations (2.3)–(2.8) are supplemented by Blasius initial conditions as $z\to 0$.

Introducing the scaled variables

$$R=r,Z=\frac{z}{Re},$$ 2.9

and

$$U_z=u_z,U_r=R{u}_r\hspace{0.17em}Re,U_{\theta }=R{u}_{\theta },P=R^{2}p,$$ 2.10

equations (2.3)–(2.8) become

$$\begin{array}{rl}& U_z\frac{\mathrm{\partial }{U}_z}{\mathrm{\partial }Z}+\frac{U_r}{R}\frac{\mathrm{\partial }{U}_z}{\mathrm{\partial }R}=-\frac{1}{R^2}\frac{\mathrm{\partial }P}{\mathrm{\partial }Z}+\frac{{\mathrm{\partial }}^2{U}_z}{\mathrm{\partial }{R}^2}+\frac{1}{R}\frac{\mathrm{\partial }{U}_z}{\mathrm{\partial }R},\end{array}$$ 2.11

$$\begin{array}{rl}& U_{\theta }^2=R\frac{\mathrm{\partial }P}{\mathrm{\partial }R}-2P,\end{array}$$ 2.12

$$\begin{array}{rl}& U_z\frac{\mathrm{\partial }{U}_{\theta }}{\mathrm{\partial }Z}+\frac{U_r}{R}\frac{\mathrm{\partial }{U}_{\theta }}{\mathrm{\partial }R}=\frac{{\mathrm{\partial }}^2{U}_{\theta }}{\mathrm{\partial }{R}^2}-\frac{1}{R}\frac{\mathrm{\partial }{U}_{\theta }}{\mathrm{\partial }R},\end{array}$$ 2.13

$$\begin{array}{rl}& \frac{\mathrm{\partial }{U}_z}{\mathrm{\partial }Z}+\frac{1}{R}\frac{\mathrm{\partial }{U}_r}{\mathrm{\partial }R}=0,\end{array}$$ 2.14

$$\begin{array}{rl}& U_z=U_r=0,U_{\theta }=SR=1\end{array}$$ 2.15

$$\begin{array}{rl}\text{and}& U_z\to 1,U_{\theta }=0,P=0R\to \mathrm{\infty }.\end{array}$$ 2.16

It is apparent that, using these scalings, Re has disappeared from the problem, leaving S as the only non-dimensional parameter. This result indicates, among other things, that the natural scaling of the axial coordinate is z=O(Re). Thus, as noted earlier, the distance needed for the boundary-layer thickness to become comparable with the radius is Re times the radius. The factors of R appearing in equation (2.10) have been introduced to improve numerical convergence.

3. Numerical scheme and validation

The boundary-layer thickness goes to zero like Z^1/2 and $U_r\to \mathrm{\infty }$ like Z^−1/2 as Z=0 is approached. To maintain numerical accuracy in the presence of such singular behaviour, we introduce the variables

$$\zeta ={(2Z)}^{1/2},\sigma =\frac{R-1}{\zeta },V_r=\zeta U_r,V_z=U_z,V_{\theta }=U_{\theta }.$$ 3.1

Here, the boundary-layer thickness is prevented from going to zero in the radial coordinate σ by dividing R−1 by ζ. V _r is kept finite by use of the factor ζ, and ζ is used in place of Z to make the solution a smooth function of the axial coordinate. Using these variables in the boundary-layer equations (2.11)–(2.16) results in

$$\begin{array}{rl}& \zeta V_z\frac{\mathrm{\partial }{V}_z}{\mathrm{\partial }\zeta }+(\frac{V_r-\zeta }{R}-\sigma V_z)\frac{\mathrm{\partial }{V}_z}{\mathrm{\partial }\sigma }=\frac{1}{R^2}(\sigma \frac{\mathrm{\partial }P}{\mathrm{\partial }\sigma }-\zeta \frac{\mathrm{\partial }P}{\mathrm{\partial }\zeta })+\frac{{\mathrm{\partial }}^2{V}_z}{\mathrm{\partial }{\sigma }^2},\end{array}$$ 3.2

$$\begin{array}{rl}& \frac{\zeta }{R}(V_{\theta }^2+2P)=\frac{\mathrm{\partial }P}{\mathrm{\partial }\sigma },\end{array}$$ 3.3

$$\begin{array}{rl}& \zeta V_z\frac{\mathrm{\partial }{V}_{\theta }}{\mathrm{\partial }\zeta }+(\frac{V_r+\zeta }{R}-\sigma V_z)\frac{\mathrm{\partial }{V}_{\theta }}{\mathrm{\partial }\sigma }=\frac{{\mathrm{\partial }}^2{V}_{\theta }}{\mathrm{\partial }{\sigma }^2}\end{array}$$ 3.4

$$\begin{array}{rl}\text{and}& \zeta \frac{\mathrm{\partial }{V}_z}{\mathrm{\partial }\zeta }-\sigma \frac{\mathrm{\partial }{V}_z}{\mathrm{\partial }\sigma }+\frac{1}{R}\frac{\mathrm{\partial }{V}_r}{\mathrm{\partial }\sigma }=0,\end{array}$$ 3.5

with the boundary conditions

$$V_z=V_r=0,V_{\theta }=S\sigma =0$$ 3.6

and

$$V_z\to 1,V_{\theta }=0,P=0\sigma \to \mathrm{\infty }.$$ 3.7

These equations govern the axial evolution of the flow. The inlet condition (Blasius solution) is obtained by setting ζ=0 and solving the resulting equations.

The radial coordinate σ is discretized using Chebyshev collocation points:

$$x_n=\cos \left(\frac{n\pi }{N-1}\right)0\le n<N$$ 3.8

and

$${\sigma }_n=\frac{\hat{\sigma }(1+x_n)}{1-x_n},x\in [-1,1]\to \sigma \in [0,\mathrm{\infty }].$$ 3.9

The parameter $\hat{\sigma }$ controls the distribution of points such that half of them lie between $0\le \sigma \le \hat{\sigma }$. The velocities V _z and V _θ are represented by their values at all collocation points. However, since there is no boundary condition for the pressure at the surface, it is represented at all points except σ=0. Similarly, there is no boundary condition for V _r at $\sigma =\mathrm{\infty }$ and so it is represented at all points apart from $\sigma =\mathrm{\infty }$. The Chebyshev derivative matrices for P and V _r are correspondingly modified (e.g. appendix A in [11]).

The coordinate ζ is discretized using small, equally spaced steps, ζ_i=iΔ, and the variables V _z,V _r,V _θ,P are represented by their values at ζ_i. Equations (3.2), (3.4) and (3.5) are evaluated at mid-step, ζ_i+1/2, using an implicit scheme that employs centred finite differencing to represent the ζ-derivatives. Equation (3.3) is evaluated at the step position ζ_i, rather than at the midstep. At each step, the equations are solved using Newton–Raphson iteration, thus allowing forward marching. The inlet solution is obtained from equations (3.2)–(3.7) using ζ=0. Following discretization in σ using the collocation points, the result is again obtained by Newton–Raphson iteration.

The code was first tested by changing the numerical parameters $N,\hat{\sigma },\mathrm{\Delta },$ and observing the dependence of the solution on these parameters. Based on the convergence results, we decided to use $N=128,\hspace{0.17em}\hat{\sigma }=5,\hspace{0.17em}\mathrm{\Delta }=0.001$ in our computations. These values gave convergence to better than seven decimal places. The code was also tested using the volume-flux and momentum balance equations. The results respect these equations to seven decimal places. Although use of the Blasius solution at the inlet has earlier been justified by an asymptotic argument, it is interesting to see the effect of a change in inlet profile on the solution. Thus, we modified the inlet profile to be $U_z^{\ast }=U_z+A\sigma \hspace{0.17em}\text{exp}(-0.5\sigma )$, where U_z is the Blasius profile. Taking A=2, it was found that the change in U_z at Z=2 was in the sixth decimal place. This illustrates the fact that the flow forgets the initial condition as Z increases and becomes insensitive to the precise inlet profile used.

We also validated the code by comparing our results with the existing literature. Tutty et al. [5] studied the case without rotation. The axial (x_t) and radial (σ_t) coordinates used by Tutty et al. [5] are related to those used here via x_t/Re=Z and ${\sigma }_t=\sqrt{2}\sigma$. Figure 2 shows good agreement with our results for Re=10⁴, x_t=0.01 and x_t=10⁵. Herrada et al. [9] considered the problem of the rotating cylinder. They do not give velocity profiles, but rather the skin friction on the cylinder:

$$\tau ={\frac{\mathrm{\partial }{U}_z(Z)}{\mathrm{\partial }R}|}_{R=1}={\frac{1}{\zeta }\frac{\mathrm{\partial }{V}_z(\zeta )}{\mathrm{\partial }\sigma }|}_{\sigma =0}.$$ 3.10

Figure 2 shows τ as a function of Z for S=1 and good agreement is apparent.

Figure 2.

Comparison of U_z for S=0 at two values of Z with [5] and comparison of skin friction τ for S=1 with [9].

4. Results

Flow profiles were obtained for different values of S and Z. Figure 3 shows velocity profiles for S=0.1 and different values of Z. For Z=0, V _z(σ) corresponds to the Blasius flat-plate solution. As Z increases, V _z(σ) deviates from the Blasius profile due to cylinder curvature and rotation. It should be borne in mind that the boundary-layer thickness increases with Z, although this is not apparent in the figure because the scaled radial coordinate σ=(R−1)/ζ has been used. Note that the azimuthal velocity at Z=0 is V _θ(σ)=S(1−V _z(σ)). As Z increases, small departures from this profile arise. Figures 4 and 5 show results for Z=0.5 and different values of S. When $S\lesssim 1$, V _z(σ) is a modified Blasius profile. However, for $S\gtrsim 4$, V _z(σ) is no longer monotonic having a maximum at finite σ. At large S, the maximum is large and the profile is better described as an axial wall jet, rather than a boundary layer. At first sight, it is perhaps surprising that increasing the rotation rate leads to a stronger and stronger axial flow. Increasing S causes V _θ to increase (figure 4). This in turn produces an increasing radial pressure gradient due to the centrifugal force. Since the pressure is constant outside the boundary layer, the pressure within the layer drops (figure 5) with Z. The development of the flow means that the axial pressure gradient becomes larger and larger, thus driving a strong axial wall jet. Whereas for lower values of S, boundary-layer thickening leads to positive V _r, at large S entrainment by the wall jet gives negative V _r outside the layer (figure 4).

Figure 3.

Velocity profiles V _z(σ), V _θ(σ) and V _r(σ) for different axial positions Z at S=0.1.

Figure 4.

Velocity profiles V _z(σ), V _θ(σ) and V _r(σ) at Z=0.5 for different values of rotation rate S.

Figure 5.

Pressure profile P(σ) at Z=0.5 for different values of S.

The existence of a wall jet at large S is apparent in the axial velocity profiles given by Petrov [7]. However, that article only gives such profiles for the case of small Z (thin boundary layer) and no explanation is provided. Petrov [7] also gives the maximum axial velocity $U_z^{max}=\underset{R}{max}(U_z(R))$ as a function of Z. Figure 6 shows a comparison with our results. A small difference is apparent, the origin of which is unclear.

Figure 6.

Comparison of $U_z^{max}$ as a function of Z^1/2 obtained by the present study with [7] for the case of S=5 and S=10.

Figure 7 shows contours of constant $U_z^{max}$ in the (S,ζ)-plane as well as the boundary (solid line) separating the region in which $U_z^{max}=1$ from that in which $U_z^{max}>1$ (which we interpret as indicating a wall jet). It will be seen that there is a threshold, S=4.15, below which $U_z^{max}=1$. Above this value, the wall jet exists for some range of axial position. Note that, whatever the strength of rotation, the wall jet eventually disappears sufficiently far downstream.

Figure 7.

Greyscale plot of $U_z^{max}$ alongside a contour plot of $U_z^{max}$. The figure also shows the boundary (solid line) between $U_z^{max}=1$ and $U_z^{max}>1$.

The thickness of the boundary layer/wall jet can be measured using

$$\delta =\frac{1}{U_z^{max}}{\int }_1^{\mathrm{\infty }}|1-U_z|\hspace{0.17em}\mathrm{d}R.$$ 4.1

The absolute value is taken to make the integral always positive and the division by $U_z^{max}$ allows for the strong wall jets which arise at large S. Figure 8 shows δ(Z) for different values of S. The layer thickness is seen to increase with Z in a roughly parabolic manner (recall that the thickness behaves as Z^1/2 for small Z). Thickening of the layer is due to viscous diffusion in the usual manner. Increasing S causes the layer to become thinner. At large S, the wall jet is of increasing strength. Viscous diffusion competes with axial convection, the latter being of growing importance, hence the decrease of δ with increasing S.

Figure 8.

Boundary-layer/wall-jet thickness δ versus Z for different values of S.

5. Asymptotic analysis

(a). Large-Z asymptotics

Suitable coordinates are

$$\eta =\frac{R}{\zeta },\chi =\ln (\zeta ).$$ 5.1

Here, we have followed Glauert & Lighthill [3], who used a logarithmic axial coordinate for the non-rotating cylinder problem. This coordinate reflects slower and slower evolution of the flow in the streamwise direction as Z increases. Using these coordinates, equations (2.11)–(2.16) become

$$\begin{array}{rl}& U_z(\frac{\mathrm{\partial }{U}_z}{\mathrm{\partial }\chi }-\eta \frac{\mathrm{\partial }{U}_z}{\mathrm{\partial }\eta })+\frac{U_r}{\eta }\frac{\mathrm{\partial }{U}_z}{\mathrm{\partial }\eta }=\frac{{\text{e}}^{-2\chi }}{{\eta }^2}(\eta \frac{\mathrm{\partial }P}{\mathrm{\partial }\eta }-\frac{\mathrm{\partial }P}{\mathrm{\partial }\chi })+\frac{{\mathrm{\partial }}^2{U}_z}{\mathrm{\partial }{\eta }^2}+\frac{1}{\eta }\frac{\mathrm{\partial }{U}_z}{\mathrm{\partial }\eta },\end{array}$$ 5.2

$$\begin{array}{rl}& U_{\theta }^2+2P=\eta \frac{\mathrm{\partial }P}{\mathrm{\partial }\eta },\end{array}$$ 5.3

$$\begin{array}{rl}& U_z(\frac{\mathrm{\partial }{U}_{\theta }}{\mathrm{\partial }\chi }-\eta \frac{\mathrm{\partial }{U}_{\theta }}{\mathrm{\partial }\eta })+\frac{U_r}{\eta }\frac{\mathrm{\partial }{U}_{\theta }}{\mathrm{\partial }\eta }=\frac{{\mathrm{\partial }}^2{U}_{\theta }}{\mathrm{\partial }{\eta }^2}-\frac{1}{\eta }\frac{\mathrm{\partial }{U}_{\theta }}{\mathrm{\partial }\eta },\end{array}$$ 5.4

$$\begin{array}{rl}& \frac{\mathrm{\partial }{U}_z}{\mathrm{\partial }\chi }-\eta \frac{\mathrm{\partial }{U}_z}{\mathrm{\partial }\eta }+\frac{1}{\eta }\frac{\mathrm{\partial }{U}_r}{\mathrm{\partial }\eta }=0,\end{array}$$ 5.5

$$\begin{array}{rl}& U_z=U_r=0,U_{\theta }=S\eta =e^{-\chi }\end{array}$$ 5.6

$$\begin{array}{rl}\text{and}& U_z\to 1,U_{\theta }\to 0,P\to 0\eta \to \mathrm{\infty }.\end{array}$$ 5.7

It is shown in appendix A that U_z,U_r,U_θ and P have asymptotic expansions in powers of χ⁻¹. The factor of e^−2χ in equation (5.2) is exponentially small and is hence negligible at all algebraic orders. Without the corresponding term in equation (5.2), U_z and U_r decouple from U_θ and P, though the latter depends on the former. Thus, we expect such decoupling to hold at all orders. This is indeed what is found in appendix A, where the governing equations for the coefficients of the expansions in powers of χ⁻¹ are obtained for all orders. Given decoupling, rotation does not enter into the asymptotics of U_z and U_r, which are consequently the same as for the non-rotating case.

Glauert & Lighthill [3] studied the case without rotation and obtained the expansions of U_z and U_r. Appendix A extends the analysis to include rotation and gives detailed results up to order 5. At first order, the asymptotic solution can be obtained analytically and is given by

$$\begin{array}{rl}& U_z\sim 1-{\chi }^{-1}{\int }_{\eta }^{\mathrm{\infty }}\frac{{\text{e}}^{-{\xi }^2/2}}{\xi }\hspace{0.17em}\mathrm{d}\xi ,\end{array}$$ 5.8

$$\begin{array}{rl}& U_r\sim {\chi }^{-1}(1-{\text{e}}^{-{\eta }^2/2}),\end{array}$$ 5.9

$$\begin{array}{rl}& U_{\theta }\sim S{\text{e}}^{-{\eta }^2/2}\end{array}$$ 5.10

$$\begin{array}{rl}\text{and}& P\sim -S^2{\eta }^2{\int }_{\eta }^{\mathrm{\infty }}\frac{{\text{e}}^{-{\xi }^2}}{{\xi }^3}\mathrm{d}\xi .\end{array}$$ 5.11

In figure 9, the results for U_z show convergence to the asymptotic form (5.8), while those for u_θ=U_θ/R converge to u_θ∼S/R, which is the flow due to a rotating cylinder, infinite in both axial directions (rather than semi-infinite) and without axial flow.

Figure 9.

Comparison of χ(U_z−1) and u_θ=U_θ/R for different values of Z at S=0.1 with the $Z\to \mathrm{\infty }$ solution.

(b). Large-S asymptotics

As we saw in the previous section, the numerical results show the existence of a wall jet at large S. In this limit, appropriate scaled variables are

$$Z^{\ast }=\frac{Z}{S},R^{\ast }=R$$ 5.12

and

$$U_z^{\ast }=\frac{U_z}{S},U_r^{\ast }=U_r,U_{\theta }^{\ast }=\frac{U_{\theta }}{S},P^{\ast }=\frac{P}{S^2}.$$ 5.13

The scaling of Z reflects the increasing distance required for flow development as the rotation rate increases. The scaling of U_z and U_θ indicates the strengthening flow velocity as S increases. The large-S asymptotic expansions of $U_z^{\ast },U_r^{\ast },U_{\theta }^{\ast }$ and P* proceed as powers of S⁻¹. At leading order (S⁰), we find

$$\begin{array}{rl}& U_z^{\ast }\frac{\mathrm{\partial }{U}_z^{\ast }}{\mathrm{\partial }{Z}^{\ast }}+\frac{U_r^{\ast }}{R^{\ast }}\frac{\mathrm{\partial }{U}_z^{\ast }}{\mathrm{\partial }{R}^{\ast }}=-\frac{1}{R^{\ast 2}}\frac{\mathrm{\partial }{P}^{\ast }}{\mathrm{\partial }{Z}^{\ast }}+\frac{{\mathrm{\partial }}^2{U}_z^{\ast }}{\mathrm{\partial }{R^{\ast }}^2}+\frac{1}{R^{\ast }}\frac{\mathrm{\partial }{U}_z^{\ast }}{\mathrm{\partial }{R}^{\ast }},\end{array}$$ 5.14

$$\begin{array}{rl}& U_{\theta }^{\ast 2}=R^{\ast }\frac{\mathrm{\partial }{P}^{\ast }}{\mathrm{\partial }{R}^{\ast }}-2P^{\ast },\end{array}$$ 5.15

$$\begin{array}{rl}& U_z^{\ast }\frac{\mathrm{\partial }{U}_{\theta }^{\ast }}{\mathrm{\partial }{Z}^{\ast }}+\frac{U_r^{\ast }}{R^{\ast }}\frac{\mathrm{\partial }{U}_{\theta }^{\ast }}{\mathrm{\partial }{R}^{\ast }}=\frac{{\mathrm{\partial }}^2{U}_{\theta }^{\ast }}{\mathrm{\partial }{R^{\ast }}^2}-\frac{1}{R^{\ast }}\frac{\mathrm{\partial }{U}_{\theta }^{\ast }}{\mathrm{\partial }{R}^{\ast }}\end{array}$$ 5.16

$$\begin{array}{rl}\text{and}& \frac{\mathrm{\partial }{U}_z^{\ast }}{\mathrm{\partial }{Z}^{\ast }}+\frac{1}{R^{\ast }}\frac{\mathrm{\partial }{U}_r^{\ast }}{\mathrm{\partial }{R}^{\ast }}=0,\end{array}$$ 5.17

with the following inlet and boundary conditions:

$$\begin{array}{rl}{U}_z^{\ast }& =0,U_{\theta }^{\ast }=0Z^{\ast }=0,\end{array}$$ 5.18

$$\begin{array}{rl}{U}_z^{\ast }& =0,U_r^{\ast }=0,U_{\theta }^{\ast }=1R^{\ast }=1\end{array}$$ 5.19

$$\begin{array}{rl}\text{and}{U}_z^{\ast }& =0,U_{\theta }^{\ast }=0,P^{\ast }=0R^{\ast }\to \mathrm{\infty }.\end{array}$$ 5.20

Figure 10 shows the solution of the above problem (solid line) compared with the numerical results discussed before for Z*=0.1 and different values of S. It is apparent that the asymptotics are indeed approached as $S\to \mathrm{\infty }$. Figure 11 shows the leading-order asymptotic solution for different values of Z*. We see that the large-Z limit (u_θ∼SR⁻¹) is approached by u_θ as $Z^{\ast }\to \mathrm{\infty }$.

Figure 10.

Comparison of $U_z^{\ast }$ and $U_{\theta }^{\ast }$ for different values of S and Z*=0.1 with the $S\to \mathrm{\infty }$ asymptotic solution.

Figure 11.

$u_z/S=U_z^{\ast }$ and u_θ/S at different values of Z* for a rotating cylinder in the limit $S\to \mathrm{\infty }$.

Note that the limit $S\to \mathrm{\infty }$ can be reached in two ways: either by increasing the rotation rate, or by decreasing the velocity $U_{\mathrm{\infty }}$ to zero. Note also that u_z/S and u_θ/S are the velocity components non-dimensionalized by Ωa, rather than $U_{\mathrm{\infty }}$, and that Z*=z/Re_Ω, where Re_Ω=Ωa²/ν is the Reynolds number based on the rotational velocity Ωa. Thus, figure 11 can be interpreted as showing the flow due to a rotating, semi-infinite cylinder in a still fluid ($U_{\mathrm{\infty }}=0$). It can be shown that the separation of radial and axial length scales, which underlies the boundary-layer type approximation we have used, is valid if either of the Reynolds numbers, Re or Re_Ω, is large.

6. Conclusion

In this paper, we have presented a study of the flow around a rotating cylinder in an axial stream. We have assumed a smooth nose to avoid flow separation. The two non-dimensional control parameters of the problem are: Reynolds number (Re) and rotation rate (S). The flow equations are formulated using a boundary-layer type approximation, appropriate at large Reynolds numbers and in which the flow is assumed to evolve slowly in the streamwise direction in comparison to the radial direction. The resulting equations are not limited to the case in which the boundary layer is thin compared with the cylinder radius. By using appropriate scalings, we remove Re from the problem.

The results show that the boundary-layer thickness increases with axial distance, becoming comparable with the cylinder radius a at distances of O(Re a). Prior to this, the layer is thin compared with the radius and the flow is close to the Blasius profile of a flat plate. However, it differs from the Blasius solution due to the effects of curvature and rotation at larger downstream distances. As S increases, the centrifugal force creates an increasing radial pressure gradient, which combined with axial development, implies an increasing axial gradient of pressure. Above S=4.15, the maximum velocity exceeds the free-stream velocity for a range of Z and we say that a wall jet exists. This jet becomes stronger and stronger as $S\to \mathrm{\infty }$.

In the limit of large Z, we find that the axial and radial components of velocity decouple from the azimuthal velocity component and pressure. All these quantities are found to have asymptotic expansions in inverse powers of ln(Z), a result already obtained for the non-rotating case by Glauert & Lighthill [3], and here extended to include rotation. The leading-order term in the u_θ expansion is u_θ∼SR⁻¹, which is the flow expected for a rotating cylinder, infinite in both directions. Because U_θ becomes independent of Z, the same is true of the pressure field resulting from the centrifugal force, hence the absence of an axial pressure gradient to drive the axial/radial flow. This is the reason for the decoupling.

When S is large, we introduce appropriate scalings for Z, U_z,U_θ and P. The asymptotic expansions of the scaled velocity and pressure proceed as inverse powers of S, beginning with S⁰. The leading-order term describes an axial wall jet due to a rotating cylinder in a fluid at rest.

Appendix A. Large Z asymptotic expansions

The flow variables are expressed as asymptotic expansions in inverse powers of χ:

$$\begin{array}{rl}{U}_z& \sim 1+\sum _{n=1}^{\mathrm{\infty }}{\chi }^{-n}{U}_z^{(n)}(\eta ),\end{array}$$ A 1

$$\begin{array}{rl}{U}_r& \sim \sum _{n=1}^{\mathrm{\infty }}{\chi }^{-n}{U}_r^{(n)}(\eta ),\end{array}$$ A 2

$$\begin{array}{rl}{U}_{\theta }& \sim S\sum _{n=1}^{\mathrm{\infty }}{\chi }^{-n+1}{U}_{\theta }^{(n)}(\eta )\end{array}$$ A 3

$$\begin{array}{rl}\text{and}P& \sim S^2\sum _{n=1}^{\mathrm{\infty }}{\chi }^{-n+1}{P}^{(n)}(\eta ).\end{array}$$ A 4

Introducing these expansions into equations (5.2)–(5.5) gives

$$\begin{array}{rl}\frac{1}{\eta }\frac{\mathrm{d}}{\mathrm{d}\eta }\left(\eta \frac{\mathrm{d}{U}_z^{(n)}}{\mathrm{d}\eta }\right)+\eta \frac{\mathrm{d}{U}_z^{(n)}}{\mathrm{d}\eta }& ={\varphi }_z^{(n)},\end{array}$$ A 5

$$\begin{array}{rl}{\eta }^3\frac{\mathrm{d}}{\mathrm{d}\eta }\left(\frac{P^{(n)}}{{\eta }^2}\right)& ={\psi }^{(n)},\end{array}$$ A 6

$$\begin{array}{rl}\eta \frac{\mathrm{d}}{\mathrm{d}\eta }\left(\frac{1}{\eta }\frac{\mathrm{d}{U}_{\theta }^{(n)}}{\mathrm{d}\eta }\right)+\eta \frac{\mathrm{d}{U}_{\theta }^{(n)}}{\mathrm{d}\eta }& ={\varphi }_{\theta }^{(n)}\end{array}$$ A 7

$$\begin{array}{rl}\text{and}\frac{1}{\eta }\frac{\mathrm{d}{U}_r^{(n)}}{\mathrm{d}\eta }-\eta \frac{\mathrm{d}{U}_z^{(n)}}{\mathrm{d}\eta }& ={\varphi }_r^{(n)},\end{array}$$ A 8

where

$$\begin{array}{rl}{\varphi }_z^{(n)}& =(1-n)U_z^{(n-1)}+\sum _m((\frac{U_r^{(m)}}{\eta }-\eta U_z^{(m)})\frac{\mathrm{d}{U}_z^{(n-m)}}{\mathrm{d}\eta }-m{U}_z^{(m)}{U}_z^{(n-m-1)}),\end{array}$$ A 9

$$\begin{array}{rl}{\varphi }_{\theta }^{(n)}& =(2-n)U_{\theta }^{(n-1)}+\sum _m((\frac{U_r^{(m)}}{\eta }-\eta U_z^{(m)})\frac{\mathrm{d}{U}_{\theta }^{(n-m)}}{\mathrm{d}\eta }-(m-1)U_{\theta }^{(m)}{U}_z^{(n-m-1)})\end{array}$$ A 10

$$\begin{array}{rl}\text{and}{\varphi }_r^{(n)}& =(n-1)U_z^{(n-1)},{\psi }^{(n)}=\sum _m{U}_{\theta }^{(m)}{U}_{\theta }^{(n-m+1)}.\end{array}$$ A 11

Equations (A 5)–(A 8) are to be solved, along with appropriate boundary conditions (which will be derived shortly), for the nth-order coefficients of the expansions, $U_z^{(n)},\hspace{0.17em}{U}_r^{(n)},\hspace{0.17em}{U}_{\theta }^{(n)}$ and P⁽ⁿ⁾ (n≥1). It should be noted that, in equations (A 9)–(A 11), $U_z^{(m)}$, $U_r^{(m)}$ and $U_{\theta }^{(m)}$ are to be interpreted as zero when m≤0. The governing equations for $U_z^{(n)}$ and $U_r^{(n)}$ are independent of $U_{\theta }^{(n)}$ and P⁽ⁿ⁾. Thus, the asymptotics of $U_z^{(n)}$ and $U_r^{(n)}$ are the same as for a non-rotating cylinder and are governed by equations (A 5), (A 8), (A 9) and the first of the equations (A 11). $U_{\theta }^{(n)}$ is determined by equations (A 7) and (A 10), while P⁽ⁿ⁾ follows from equation (A 6) and the second of the equations (A 11). Note that ${\varphi }_z^{(n)},{\varphi }_r^{(n)}$ and ${\varphi }_{\theta }^{(n)}$ depend only on the solution at lower orders than n, suggesting a method which proceeds from n=1 to successively higher values of n.

The boundary conditions at $\eta \to \mathrm{\infty }$ are

$$U_z^{(n)}=U_{\theta }^{(n)}=P^{(n)}=0.$$ A 12

Application of the boundary conditions (2.15) at the cylinder surface requires the introduction of an inner region, R=O(1), represented by the expansions

$$\begin{array}{rl}{U}_z& \sim \sum _{n=1}^{\mathrm{\infty }}{\chi }^{-n}{\hat{U}}_z^{(n)}(R),\end{array}$$ A 13

$$\begin{array}{rl}{U}_r& \sim \sum _{n=1}^{\mathrm{\infty }}{\chi }^{-n}{\hat{U}}_r^{(n)}(R),\end{array}$$ A 14

$$\begin{array}{rl}{U}_{\theta }& \sim S\sum _{n=1}^{\mathrm{\infty }}{\chi }^{-n+1}{\hat{U}}_{\theta }^{(n)}(R)\end{array}$$ A 15

$$\begin{array}{rl}\text{and}P& \sim S^2\sum _{n=1}^{\mathrm{\infty }}{\chi }^{-n+1}{\hat{P}}^{(n)}(R).\end{array}$$ A 16

Equations (2.11), (2.13) and (2.14) are rewritten using the axial coordinate χ in place of Z. Equation (2.14) yields

$$\frac{\mathrm{\partial }{\hat{U}}_r^{(n)}}{\mathrm{\partial }R}=0,$$ A 17

which, together with the boundary conditions (2.15) gives ${\hat{U}}_r^{(n)}=0$. Equations (2.11) and (2.13) imply

$$\frac{{\mathrm{\partial }}^2{\hat{U}}_z^{(n)}}{\mathrm{\partial }{R}^2}+\frac{1}{R}\frac{\mathrm{\partial }{\hat{U}}_z^{(n)}}{\mathrm{\partial }R}=0$$ A 18

and

$$\frac{{\mathrm{\partial }}^2{\hat{U}}_{\theta }^{(n)}}{\mathrm{\partial }{R}^2}-\frac{1}{R}\frac{\mathrm{\partial }{\hat{U}}_{\theta }^{(n)}}{\mathrm{\partial }R}=0,$$ A 19

hence

$${\hat{U}}_z^{(n)}(R)=A_n\ln R+B_n$$ A 20

and

$${\hat{U}}_{\theta }^{(n)}(R)=C_n{R}^2+D_n.$$ A 21

The boundary conditions (2.15) imply B_n=0, C₁+D₁=1, and C_n+D_n=0 for n>1.

Recalling that R=ζη=e^χη, the inner expansions give

$$\begin{array}{rl}{U}_z& \sim A_1+\sum _{n=1}^{\mathrm{\infty }}{\chi }^{-n}(A_n\ln \eta +A_{n+1}),\end{array}$$ A 22

$$\begin{array}{rl}{U}_{\theta }& \sim S(1+\sum _{n=1}^{\mathrm{\infty }}{C}_n{\chi }^{-n}({\mathrm{e}}^{2\chi }{\eta }^2-1))\end{array}$$ A 23

$$\begin{array}{rl}\text{and}{U}_r& \sim 0,\end{array}$$ A 24

when expressed in terms of the outer coordinate, η. Matching requires A₁=1, C_n=0 and

$$\begin{array}{rl}& U_z^{(n)}\sim A_n\ln \eta +A_{n+1},\end{array}$$ A 25

$$\begin{array}{rl}& U_{\theta }^{(1)}\to 1\text{and}{U}_{\theta }^{(n)}\to 0\hspace{0.17em}\text{for}n>1\end{array}$$ A 26

and

$$U_r^{(n)}\to 0,$$ A 27

as $\eta \to 0$. It follows from (A 25) that

$$\eta \frac{\mathrm{d}{U}_z^{(n)}}{\mathrm{d}\eta }\to A_n$$ A 28

and

$$A_{n+1}=\underset{\eta \to 0}{lim}(U_z^{(n)}-A_n\text{ln}\eta ).$$ A 29

Assuming A_n is known, equations (A 5)–(A 11) and the boundary conditions (A 12) and (A 26)–(A 28) can be solved for $U_z^{(n)}$, $U_r^{(n)}$, $U_{\theta }^{(n)}$ and P⁽ⁿ⁾. A₁=1 gets the process started and leads to the leading-order outer solution, (5.8)–(5.11), in agreement with [3]. Equation (A 29) gives A_n at the next order, allowing solution at successively higher orders. It can be shown that

$$U_z^{(n)}=A_n\hspace{0.17em}\text{ln}\eta +A_{n+1}+O({\eta }^2\hspace{0.17em}{\text{ln}}^{p_n}\eta ),U_{\theta }^{(n)}=1-q_n+O({\eta }^2),U_r^{(n)}=O({\eta }^2\hspace{0.17em}{\text{ln}}^{q_n}\eta ),$$ A 30

as $\eta \to 0$, where q_n=p_n+1−1, p₁=0, p₂=1 and p_n=2 for n≥3. The terms in (A 30) indicated by the O() notation are exponentially small in the inner region, while the remaining ones reproduce the inner solution. Thus, the outer expansions in fact apply in the inner region.

The above procedure has been implemented numerically and results up to n=5 are presented in figure 12. Figure 13 shows the comparison of numerical solution of U_z at Z=5000 and S=1 with the asymptotic solution obtained by truncating at different orders n. Although this result shows good convergence, and therefore further confirms both numerical and analytical results, it should be borne in mind that the expansions (A 1)–(A 4) are, in fact, asymptotic as $Z\to \mathrm{\infty }$, rather than necessarily convergent at any finite Z.

Figure 12.

First five coefficients of the large-Z asymptotic expansions of U_z, U_θ and U_r.

Figure 13.

Comparison of the numerical solution of U_z with the large-Z asymptotic solution truncated at different orders for Z=5000 at S=1.

Appendix B. Flow over the nose

Large Reynolds number implies a thin boundary layer over the nose. Schlichting [12] gives the axisymmetric boundary-layer equations in terms of curvilinear coordinates, x,y,θ, where x is distance along the surface, and y is distance normal to the surface. Here, we use the non-dimensional coordinates, velocity components and pressure:

$$\tilde{y}=\frac{{Re}^{1/2}y}{a},\tilde{x}=\frac{x}{a}$$ B 1

and

$${\tilde{u}}_y=\frac{{Re}^{1/2}{u}_y}{U_{\mathrm{\infty }}},{\tilde{u}}_x=\frac{u_x}{U_{\mathrm{\infty }}},{\tilde{u}}_{\theta }=\frac{u_{\theta }}{U_{\mathrm{\infty }}},\tilde{p}=p.$$ B 2

The boundary-layer equations in these variables are

$$\begin{array}{rl}{\tilde{u}}_x\frac{\mathrm{\partial }{\tilde{u}}_x}{\mathrm{\partial }\tilde{x}}+{\tilde{u}}_y\frac{\mathrm{\partial }{\tilde{u}}_x}{\mathrm{\partial }\tilde{y}}-\frac{{\tilde{u}}_{\theta }^2}{\tilde{R}}\frac{\mathrm{d}\tilde{R}}{\mathrm{d}\tilde{x}}& =-\frac{\mathrm{\partial }\tilde{p}}{\mathrm{\partial }\tilde{x}}+\frac{{\mathrm{\partial }}^2{\tilde{u}}_x}{\mathrm{\partial }{\tilde{y}}^2},\end{array}$$ B 3

$$\begin{array}{rl}{\tilde{u}}_x\frac{\mathrm{\partial }{\tilde{u}}_{\theta }}{\mathrm{\partial }\tilde{x}}+{\tilde{u}}_y\frac{\mathrm{\partial }{\tilde{u}}_{\theta }}{\mathrm{\partial }\tilde{y}}+\frac{{\tilde{u}}_{\theta }{\tilde{u}}_x}{\tilde{R}}\frac{\mathrm{d}\tilde{R}}{\mathrm{d}\tilde{x}}& =\frac{{\mathrm{\partial }}^2{\tilde{u}}_{\theta }}{\mathrm{\partial }{\tilde{y}}^2}\end{array}$$ B 4

$$\begin{array}{rl}\frac{\mathrm{\partial }\tilde{p}}{\mathrm{\partial }\tilde{y}}& =0\\ \text{and}\frac{\mathrm{\partial }{\tilde{u}}_x}{\mathrm{\partial }\tilde{x}}+\frac{{\tilde{u}}_x}{\tilde{R}}\frac{d\tilde{R}}{d\tilde{x}}+\frac{\mathrm{\partial }{\tilde{u}}_y}{\mathrm{\partial }\tilde{y}}& =0,\end{array}$$ B 5

with the boundary conditions

$${\tilde{u}}_x={\tilde{u}}_y=0,{\tilde{u}}_{\theta }=S\tilde{R}(\tilde{x})\tilde{y}=0$$ B 6

and

$${\tilde{u}}_x\to U_{\mathrm{ext}}(\tilde{x}),{\tilde{u}}_{\theta }\to 0\tilde{y}\to \mathrm{\infty },$$ B 7

where the nose geometry is represented by $r=\tilde{R}(\tilde{x})$ and $U_{\mathrm{ext}}(\tilde{x})$ is the velocity just outside the boundary layer. Equations (A 3)–(B 5) can, in principle, be solved to obtain the flow over the nose. Note the centrifugal term in equation (B 3), which will no doubt produce a wall jet on the nose at sufficiently large S. The terms containing $\mathrm{d}\tilde{R}/\mathrm{d}\tilde{x}$ vanish on the constant-radius cylinder (where $\tilde{x}$ and z coincide to within an additive constant) and equations (B 3)–(B 5) then become those of a flat-plate. Thus, we expect the flow to approach the Blasius solution as $\tilde{x}\to \mathrm{\infty }$. There are, in fact, two asymptotic regions, $\tilde{x}=O(1)$ and $\tilde{x}=O(Re)$, the former being described by equations (B 3)–(B 5) and the latter by equations (2.11)–(2.14). Matching of the regions requires the Blasius flow as inlet conditions to the latter equations, as noted in the main text. Thus, the flow in the region Z=O(1), which is the subject of this paper, is insensitive to the geometry of the nose. Note that a wall jet may appear on the nose, subsequently disappearing on the cylinder, later reappearing in the region Z=O(1).

Data accessibility

No new data were collected in the course of this research.

Authors' contributions

S.D.M. has carried out the research as part of his PhD, supervised by B.P. and J.F.S.; B.P. and R.G. have formulated the problem as part of the longer term Franco-Indian collaboration; S.D.M. has started on this work with R.G.

Competing interests

We have no competing interests.

Funding

The authors are grateful to the French Ministry of Education and Research and the Indo-French Centre for the Promotion of Advanced Research-CEFIPRA for financial support.