Jeffery–Hamel flow

In fluid dynamics Jeffery–Hamel flow is a flow created by a converging or diverging channel with a source or sink of fluid volume at the point of intersection of the two plane walls. It is named after George Barker Jeffery(1915)^[1] and Georg Hamel(1917),^[2] but it has subsequently been studied by many major scientists such as von Kármán and Levi-Civita,^[3] Walter Tollmien,^[4] F. Noether,^[5] W.R. Dean,^[6] Rosenhead,^[7] Landau,^[8] G.K. Batchelor^[9] etc. A complete set of solutions was described by Edward Fraenkel in 1962.^[10]

Flow description

Consider two stationary plane walls with a constant volume flow rate $Q$ is injected/sucked at the point of intersection of plane walls and let the angle subtended by two walls be $2\alpha$ . Take the cylindrical coordinate $(r,\theta ,z)$ system with $r=0$ representing point of intersection and $\theta =0$ the centerline and $(u,v,w)$ are the corresponding velocity components. The resulting flow is two-dimensional if the plates are infinitely long in the axial $z$ direction, or the plates are longer but finite, if one were neglect edge effects and for the same reason the flow can be assumed to be entirely radial i.e., $u=u(r,\theta ),v=0,w=0$ .

Then the continuity equation and the incompressible Navier–Stokes equations reduce to

{\begin{aligned}{\frac {\partial (ru)}{\partial r}}&=0,\\[6pt]u{\frac {\partial u}{\partial r}}&=-{\frac {1}{\rho }}{\frac {\partial p}{\partial r}}+\nu \left[{\frac {1}{r}}{\frac {\partial }{\partial r}}\left(r{\frac {\partial u}{\partial r}}\right)+{\frac {1}{r^{2}}}{\frac {\partial ^{2}u}{\partial \theta ^{2}}}-{\frac {u}{r^{2}}}\right]\\[6pt]0&=-{\frac {1}{\rho r}}{\frac {\partial p}{\partial \theta }}+{\frac {2\nu }{r^{2}}}{\frac {\partial u}{\partial \theta }}\end{aligned}}

The boundary conditions are no-slip condition at both walls and the third condition is derived from the fact that the volume flux injected/sucked at the point of intersection is constant across a surface at any radius.

u(\pm \alpha )=0,\quad Q=\int _{-\alpha }^{\alpha }ur\,d\theta

Formulation

The first equation tells that $ru$ is just function of $\theta$ , the function is defined as

F(\theta )={\frac {ru}{\nu }}.

Different authors defines the function differently, for example, Landau^[8] defines the function with a factor $6$ . But following Whitham,^[11] Rosenhead^[12] the $\theta$ momentum equation becomes

{\frac {1}{\rho }}{\frac {\partial p}{\partial \theta }}={\frac {2\nu ^{2}}{r^{2}}}{\frac {dF}{d\theta }}

Now letting

{\frac {p-p_{\infty }}{\rho }}={\frac {\nu ^{2}}{r^{2}}}P(\theta ),

the $r$ and $\theta$ momentum equations reduce to

P=-{\frac {1}{2}}(F^{2}+F'')

P'=2F',\quad \Rightarrow \quad P=2F+C

and substituting this into the previous equation(to eliminate pressure) results in

F''+F^{2}+4F+2C=0

Multiplying by $F'$ and integrating once,

{\frac {1}{2}}F'^{2}+{\frac {1}{3}}F^{3}+2F^{2}+2CF=D,

{\frac {1}{2}}F'^{2}+{\frac {1}{3}}(F^{3}+6F^{2}+6CF-3D)=0

where $C,D$ are constants to be determined from the boundary conditions. The above equation can be re-written conveniently with three other constants $a,b,c$ as roots of a cubic polynomial, with only two constants being arbitrary, the third constant is always obtained from other two because sum of the roots is $a+b+c=-6$ .

{\frac {1}{2}}F'^{2}+{\frac {1}{3}}(F-a)(F-b)(F-c)=0,

{\frac {1}{2}}F'^{2}-{\frac {1}{3}}(a-F)(F-b)(F-c)=0.

The boundary conditions reduce to

F(\pm \alpha )=0,\quad {\frac {Q}{\nu }}=\int _{-\alpha }^{\alpha }F\,d\theta

where $Re=Q/\nu$ is the corresponding Reynolds number. The solution can be expressed in terms of elliptic functions. For convergent flow $Q<0$ , the solution exists for all $Re$ , but for the divergent flow $Q>0$ , the solution exists only for a particular range of $Re$ .

Dynamical interpretation

Source:^[13]

The equation takes the same form as an undamped nonlinear oscillator(with cubic potential) one can pretend that $\theta$ is time, $F$ is displacement and $F'$ is velocity of a particle with unit mass, then the equation represents the energy equation( $K.E.+P.E.=0$ , where $K.E.={\frac {1}{2}}F'^{2}$ and $P.E.=V(F)$ ) with zero total energy, then it is easy to see that the potential energy is

V(F)=-{\frac {1}{3}}(a-F)(F-b)(F-c)

where $V\leq 0$ in motion. Since the particle starts at $F=0$ for $\theta =-\alpha$ and ends at $F=0$ for $\theta =\alpha$ , there are two cases to be considered.

First case $b,c$ are complex conjugates and $a>0$ . The particle starts at $F=0$ with finite positive velocity and attains $F=a$ where its velocity is $F'=0$ and acceleration is $F''=-dV/dF<0$ and returns to $F=0$ at final time. The particle motion $0<F<a$ represents pure outflow motion because $F>0$ and also it is symmetric about $\theta =0$ .
Second case $c<b<0<a$ , all constants are real. The motion from $F=0$ to $F=a$ to $F=0$ represents a pure symmetric outflow as in the previous case. And the motion $F=0$ to $F=b$ to $F=0$ with $F<0$ for all time( $-\alpha \leq \theta \leq \alpha$ ) represents a pure symmetric inflow. But also, the particle may oscillate between $b\leq F\leq a$ , representing both inflow and outflow regions and the flow is no longer need to symmetric about $\theta =0$ .

The rich structure of this dynamical interpretation can be found in Rosenhead(1940).^[7]

Pure outflow

For pure outflow, since $F=a$ at $\theta =0$ , integration of governing equation gives

\theta ={\sqrt {\frac {3}{2}}}\int _{F}^{a}{\frac {dF}{\sqrt {(a-F)(F-b)(F-c))}}}

and the boundary conditions becomes

\alpha ={\sqrt {\frac {3}{2}}}\int _{0}^{a}{\frac {dF}{\sqrt {(a-F)(F-b)(F-c))}}},\quad Re=2{\sqrt {\frac {3}{2}}}\int _{0}^{\alpha }{\frac {FdF}{\sqrt {(a-F)(F-b)(F-c))}}}.

The equations can be simplified by standard transformations given for example in Jeffreys.^[14]

First case $b,c$ are complex conjugates and $a>0$ leads to

F(\theta )=a-{\frac {3M^{2}}{2}}{\frac {1-\operatorname {cn} (M\theta ,\kappa )}{1+\operatorname {cn} (M\theta ,\kappa )}}

M^{2}={\frac {2}{3}}{\sqrt {(a-b)(a-c)}},\quad \kappa ^{2}={\frac {1}{2}}+{\frac {a+2}{2M^{2}}}

where $\operatorname {sn} ,\operatorname {cn}$ are Jacobi elliptic functions.

Second case $c<b<0<a$ leads to

F(\theta )=a-6k^{2}m^{2}\operatorname {sn} ^{2}(m\theta ,k)

m^{2}={\frac {1}{6}}(a-c),\quad k^{2}={\frac {a-b}{a-c}}.

Limiting form

The limiting condition is obtained by noting that pure outflow is impossible when $F'(\pm \alpha )=0$ , which implies $b=0$ from the governing equation. Thus beyond this critical conditions, no solution exists. The critical angle $\alpha _{c}$ is given by

{\begin{aligned}\alpha _{c}&={\sqrt {\frac {3}{2}}}\int _{0}^{a}{\frac {dF}{\sqrt {F(a-F)(F+a+6))}}},\\&={\sqrt {\frac {3}{2a}}}\int _{0}^{1}{\frac {dt}{\sqrt {t(1-t)\{1+(1+6/a)t\}}}},\\&={\frac {K(k^{2})}{m^{2}}}\end{aligned}}

where

m^{2}={\frac {3+a}{3}},\quad k^{2}={\frac {1}{2}}\left({\frac {a}{3+a}}\right)

where $K(k^{2})$ is the complete elliptic integral of the first kind. For large values of $a$ , the critical angle becomes $\alpha _{c}={\sqrt {\frac {3}{a}}}K\left({\frac {1}{2}}\right)={\frac {3.211}{\sqrt {a}}}$ .

The corresponding critical Reynolds number or volume flux is given by

{\begin{aligned}Re_{c}={\frac {Q_{c}}{\nu }}&=2\int _{0}^{\alpha _{c}}(a-6k^{2}m^{2}\operatorname {sn} ^{2}m\theta )\,d\theta ,\\&={\frac {12k^{2}}{\sqrt {1-2k^{2}}}}\int _{0}^{K}\operatorname {cn} ^{2}tdt,\\&={\frac {12}{\sqrt {1-2k^{2}}}}[E(k^{2})-(1-k^{2})K(k^{2})]\end{aligned}}

where $E(k^{2})$ is the complete elliptic integral of the second kind. For large values of $a,\left(\ k^{2}\sim {\frac {1}{2}}-{\frac {3}{2a}}\right)$ , the critical Reynolds number or volume flux becomes $Re_{c}={\frac {Q_{c}}{\nu }}=12{\sqrt {\frac {a}{3}}}\left[E\left({\frac {1}{2}}\right)-{\frac {1}{2}}K\left({\frac {1}{2}}\right)\right]=2.934{\sqrt {a}}$ .

Pure inflow

For pure inflow, the implicit solution is given by

\theta ={\sqrt {\frac {3}{2}}}\int _{b}^{F}{\frac {dF}{\sqrt {(a-F)(F-b)(F-c))}}}

and the boundary conditions becomes

\alpha ={\sqrt {\frac {3}{2}}}\int _{b}^{0}{\frac {dF}{\sqrt {(a-F)(F-b)(F-c))}}},\quad Re=2{\sqrt {\frac {3}{2}}}\int _{\alpha }^{0}{\frac {FdF}{\sqrt {(a-F)(F-b)(F-c))}}}.

Pure inflow is possible only when all constants are real $c<b<0<a$ and the solution is given by

F(\theta )=a-6k^{2}m^{2}\operatorname {sn} ^{2}(K-m\theta ,k)=b+6k^{2}m^{2}\operatorname {cn} ^{2}(K-m\theta ,k)

m^{2}={\frac {1}{6}}(a-c),\quad k^{2}={\frac {a-b}{a-c}}

where $K(k^{2})$ is the complete elliptic integral of the first kind.

Limiting form

As Reynolds number increases ( $-b$ becomes larger), the flow tends to become uniform(thus approaching potential flow solution), except for boundary layers near the walls. Since $m$ is large and $\alpha$ is given, it is clear from the solution that $K$ must be large, therefore $k\sim 1$ . But when $k\approx 1$ , $\operatorname {sn} t\approx \tanh t,\ c\approx b,\ a\approx -2b$ , the solution becomes

F(\theta )=b\left\{3\tanh ^{2}\left[{\sqrt {-{\frac {b}{2}}}}(\alpha -\theta )+\tanh ^{-1}{\sqrt {\frac {2}{3}}}\right]-2\right\}.

It is clear that $F\approx b$ everywhere except in the boundary layer of thickness $O\left({\sqrt {-{\frac {b}{2}}}}\right)$ . The volume flux is $Q/\nu \approx 2\alpha b$ so that $|Re|=O(|b|)$ and the boundary layers have classical thickness $O\left(|Re|^{1/2}\right)$ .

References

^ Jeffery, G. B. "L. The two-dimensional steady motion of a viscous fluid." The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science 29.172 (1915): 455–465.
^ Hamel, Georg. "Spiralförmige Bewegungen zäher Flüssigkeiten." Jahresbericht der Deutschen Mathematiker-Vereinigung 25 (1917): 34–60.
^ von Kármán, and Levi-Civita. "Vorträge aus dem Gebiete der Hydro-und Aerodynamik." (1922)
^ Walter Tollmien "Handbuch der Experimentalphysik, Vol. 4." (1931): 257.
^ Fritz Noether "Handbuch der physikalischen und technischen Mechanik, Vol. 5." Leipzig, JA Barch (1931): 733.
^ Dean, W. R. "LXXII. Note on the divergent flow of fluid." The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science 18.121 (1934): 759–777.
^ ^a ^b Louis Rosenhead "The steady two-dimensional radial flow of viscous fluid between two inclined plane walls." Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences. Vol. 175. No. 963. The Royal Society, 1940.
^ ^a ^b Lev Landau, and E. M. Lifshitz. "Fluid Mechanics Pergamon." New York 61 (1959).
^ G.K. Batchelor. An introduction to fluid dynamics. Cambridge university press, 2000.
^ Fraenkel, L. E. (1962). Laminar flow in symmetrical channels with slightly curved walls, I. On the Jeffery-Hamel solutions for flow between plane walls. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences, 267(1328), 119-138.
^ Whitham, G. B. "Chapter III in Laminar Boundary Layers." (1963): 122.
^ Rosenhead, Louis, ed. Laminar boundary layers. Clarendon Press, 1963.
^ Drazin, Philip G., and Norman Riley. The Navier–Stokes equations: a classification of flows and exact solutions. No. 334. Cambridge University Press, 2006.
^ Jeffreys, Harold, Bertha Swirles, and Philip M. Morse. "Methods of mathematical physics." (1956): 32–34.

[1] Jeffery, G. B. "L. The two-dimensional steady motion of a viscous fluid." The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science 29.172 (1915): 455–465.

[2] Hamel, Georg. "Spiralförmige Bewegungen zäher Flüssigkeiten." Jahresbericht der Deutschen Mathematiker-Vereinigung 25 (1917): 34–60.

[3] von Kármán, and Levi-Civita. "Vorträge aus dem Gebiete der Hydro-und Aerodynamik." (1922)

[4] Walter Tollmien "Handbuch der Experimentalphysik, Vol. 4." (1931): 257.

[5] Fritz Noether "Handbuch der physikalischen und technischen Mechanik, Vol. 5." Leipzig, JA Barch (1931): 733.

[6] Dean, W. R. "LXXII. Note on the divergent flow of fluid." The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science 18.121 (1934): 759–777.

[Rosenhead_1940-7] Louis Rosenhead "The steady two-dimensional radial flow of viscous fluid between two inclined plane walls." Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences. Vol. 175. No. 963. The Royal Society, 1940.

[Landau_Lifshitz_1959-8] Lev Landau, and E. M. Lifshitz. "Fluid Mechanics Pergamon." New York 61 (1959).

[9] G.K. Batchelor. An introduction to fluid dynamics. Cambridge university press, 2000.

[10] Fraenkel, L. E. (1962). Laminar flow in symmetrical channels with slightly curved walls, I. On the Jeffery-Hamel solutions for flow between plane walls. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences, 267(1328), 119-138.

[11] Whitham, G. B. "Chapter III in Laminar Boundary Layers." (1963): 122.

[12] Rosenhead, Louis, ed. Laminar boundary layers. Clarendon Press, 1963.

[13] Drazin, Philip G., and Norman Riley. The Navier–Stokes equations: a classification of flows and exact solutions. No. 334. Cambridge University Press, 2006.

[14] Jeffreys, Harold, Bertha Swirles, and Philip M. Morse. "Methods of mathematical physics." (1956): 32–34.

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Fahrer / Team für die Suche eingeben:

Fahrer / Team für die Suche eingeben:

Langbahn Team – Weltmeisterschaft

Jeffery–Hamel flow

Flow description

Formulation

Dynamical interpretation

Pure outflow

Limiting form

Pure inflow

Limiting form

References