HEAT TRANSFER IN AXISYMMETRIC STAGNATION FLOW ON A THIN CYLINDER

CORNELIA REVNIC, TEODOR GROŞAN, AND IOAN POP

Abstract

The steady axisymetric stagnation flow and heat transfer on a thin infinite cylinder of radius $a$ is studied in this paper. Both cases of constant wall temperature and constant wall heat flux are considered. Using similarity variables the governing partial differential equations are transformed into ordinary differential equations. The resulting set of two equations is solved numerically using Runge-Kutta method combined with a shooting technique. For the special case of the Reynolds number $Re\gg 1$ (boundary layer approximation), we obtained an asymptotic solution which include the Hiemenz solution. The present results are compared in some particular cases with existing results from the open literature and with the asymptotic approximation, and we found a very good agreement. It is shown that the Nusselt number and the skin friction increase and the boundary layer thickness decreases with the increase of the Reynolds number. Some graphs for the velocity and temperature profiles are presented. Also, tables with values related to the skin friction and Nusselt number are given.

1. Introduction

The two-dimensional orthogonal stagnation-point flow of a viscous fluid impinging on a flat wall is a very interesting problem in the history of fluid mechanics. This flow appears in virtually all flow fields of engineering and scientific interest. Hiemenz [1] was the first who derived an exact solution of the Navier-Stokes equations

⁰⁰footnotetext: Received by the editors: 01.10.2007.
2000 Mathematics Subject Classification. 76D05,80A20.
Key words and phrases. viscous and incompressible fluid, heat transfer, boundary layer.
The work has been supported by MEdC under Grant PN-II-ID-PCE-2007-1/525.

which describes the steady forced convection flow directed perpendicular (orthogonal) to an infinite flat plate. Homann [2] studied the axisymmetric stagnation flow, also against a plate, and Howarth [3] and Davey [4] extended the results to unsymmetric cases. Later, Wang [5] presented an exact solution for the steady axisymmetric stagnation-point flow on an infinite thin circular cylinder. Gorla [6] has then considered the steady boundary layer heat transfer in an axisymmetric stagnation-point flow on an infinite thin circular cylinder. Both the cases of constant wall temperature and constant wall heat flux at the surface of the cylinder were considered. Numerical results for the velocity and temperature profiles as well as for the local Nusselt number were obtained when the Reynolds number is relatively small. Further, Gorla [7] has investigated the unsteady fluid dynamic characteristics of an axisymmetric stagnation point flow on a circular cylinder performing an harmonic motion in its own plane. Also, Gorla [8] has investigated the final approach to steady state in an axisymmetric stagnation-point flow on a thin circular cylinder.

The aim of this paper is to extend the paper by Gorla [6] on heat transfer in axisymmetric stagnation point flow on a thin infinite circular cylinder to the case when the Reynolds number is large.

2. Basic equations

Consider the steady-state flow and heat transfer at an axisymmetric stagnation point on a thin circular cylinder of radius $a$ placed in a viscous and incompressible fluid of ambient uniform temperature $T_{\infty}$ , as shown in Fig. 1. The flow is axisymmetric about $z$ - axis and also symmetric to the $z=0$ plane. It is assumed that both the temperature of the surface of the cylinder $T_{w}$ or the heat flux from the surface of the cylinder $q_{w}$ are constants. Under these assumptions, the basic equations in cylindrical co-ordinates $(r,z)$ are:

Continuity

\frac{1}{r}\frac{\partial(ru)}{\partial r}+\frac{\partial w}{\partial z}=0

(1)

Navier Stokes

	$\displaystyle u\frac{\partial u}{\partial r}+w\frac{\partial u}{\partial z}=-\frac{1}{\rho}\frac{\partial p}{\partial r}+v\left(\frac{\partial^{2}u}{\partial r^{2}}+\frac{1}{r}\frac{\partial u}{\partial r}+\frac{\partial^{2}u}{\partial z^{2}}-\frac{u}{r^{2}}\right)$		(2)
	$\displaystyle u\frac{\partial w}{\partial r}+w\frac{\partial w}{\partial z}=-\frac{1}{\rho}\frac{\partial p}{\partial z}+\nu\left(\frac{\partial^{2}w}{\partial r^{2}}+\frac{1}{r}\frac{\partial w}{\partial r}+\frac{\partial^{2}w}{\partial z^{2}}\right)$		(3)

Energy

u\frac{\partial T}{\partial r}+w\frac{\partial T}{\partial z}=\alpha\left(\frac{\partial^{2}T}{\partial r^{2}}+\frac{1}{r}\frac{\partial T}{\partial r}+\frac{\partial^{2}T}{\partial z^{2}}\right)

(4)

subject to the boundary conditions of these equations

	$\displaystyle r=a:\quad u=w=0$		(5)
	$\displaystyle\quad T=T_{w}(\mathrm{CWT})\text{ or }\frac{\partial T}{\partial r}=-\frac{q_{w}}{k}(\mathrm{CWHF})$
	$\displaystyle r\longrightarrow\infty:\quad u=-A\left(r-\frac{a^{2}}{r}\right),w=2Az$
	$\displaystyle\quad T=T_{\infty}$

Here $u$ and $v$ are the velocity components along $r$ - and $z-$ axes, $T$ is the fluid temperature, $p$ is the pressure, $\rho$ is the density, $\alpha$ is the thermal expansion coefficient, $\nu$ is the kinematic viscosity and $A$ is a given constant.

In order to solve Eqs. (1) - (4), we introduce the following similarity variables

	$\displaystyle u$	$\displaystyle=-Aa\eta^{-1/2}f(\eta),w=2Af^{\prime}(\eta)z,\eta=\left(\frac{r}{a}\right)^{2},$		(6)
	$\displaystyle\theta(\eta)$	$\displaystyle=\frac{T-T_{\infty}}{T_{w}-T_{\infty}}(\mathrm{CWT}),\quad\theta(\eta)=\frac{2\left(T-T_{\infty}\right)}{\left(aq_{w}/k\right)}(\mathrm{CWHF})$

Refer to caption — Figure 1: Figure 1. The coordinate axis

Substituting (6) into Eqs. (2) and (4), we get the following ordinary differential equations

	$\displaystyle\eta f^{\prime\prime\prime}+f^{\prime\prime}+\operatorname{Re}\left(1+ff^{\prime\prime}-f^{\prime 2}\right)=0$		(7)
	$\displaystyle\left(\eta\theta^{\prime}\right)^{\prime}+\operatorname{Pr}\operatorname{Re}f\theta^{\prime}=0$		(8)

subject to the boundary conditions (5) which become

	$\displaystyle f(1)=0,f^{\prime}(1)=0,f^{\prime}(\infty)=1$		(9)
	$\displaystyle\theta(1)=1,\theta(\infty)=0(\mathrm{CWT})$
	$\displaystyle\theta^{\prime}(1)=-1,\theta(\infty)=0(\mathrm{CWHF})$

where $Re$ is the Reynolds number and $\operatorname{Pr}$ is the Prandtl number which are defined

\operatorname{Re}=\frac{Aa^{2}}{2\nu},\operatorname{Pr}=\frac{\nu}{\alpha}

(10)

The physical quantities of interest in this problem are the skin friction coefficient $C_{f}$ , the Nusselt numbers for the wall constant temperature case $Nu$ and for the constant wall heat flux case $Nu^{*}$ . It is easily to show that these quantities can be expressed as

\operatorname{Re}C_{f}=-f^{\prime\prime}(1),Nu=-2\theta^{\prime}(1)(\mathrm{CWT}),Nu^{*}=\frac{2}{\theta(1)}(\mathrm{CHF})

(11)

Case $Re\gg 1$
We consider now the boundary layer approximation $(Re\gg 1)$ of the problem under consideration. In this respect, we introduce the following new variables:

	$\displaystyle\xi$	$\displaystyle=\operatorname{Re}^{1/2}(\eta-1),f(\eta)=\operatorname{Re}^{-1/2}F(\xi)$		(12)
	$\displaystyle\theta(\eta)$	$\displaystyle=\Theta(\xi)(CWT),\theta(\eta)=\operatorname{Re}^{-1/2}\Theta(\xi)(\mathrm{CHF})$

Substituting (12) into Eqs. (7) and (8), we obtain:

	$\displaystyle\left(1+\mathrm{Re}^{-1/2}\xi\right)F^{\prime\prime\prime}+1+FF^{\prime\prime}-F^{2}+\mathrm{Re}^{-1/2}F^{\prime\prime}=0$		(13)
	$\displaystyle\left(1+\mathrm{Re}^{-1/2}\xi\right)\Theta^{\prime\prime}+\operatorname{Pr}F\Theta^{\prime}+\mathrm{Re}^{-1/2}\Theta^{\prime}=0$		(14)

along with the boundary conditions

$\displaystyle F(0)$	$\displaystyle=0,F^{\prime}(0)=0,F^{\prime}(\infty)=1$	(15)
$\displaystyle\Theta(0)$	$\displaystyle=1,\Theta(\infty)=0(CWT)$
$\displaystyle\Theta^{\prime}(0)$	$\displaystyle=-1,\Theta_{0}(\infty)=0(CWHF)$

We notice that for $Re\longrightarrow\infty$ , that corresponds to the boundary layer approximation, Eq. (13) - (15) reduce to the Hiemenz equations that describe the stagnation point flow on a plate, see Hiemenz [1]. Equations (13) - (15) were solved analytically using the following series expansions:

	$\displaystyle F=F_{0}+\operatorname{Re}^{-1/2}F_{1}+\operatorname{Re}^{-1}F_{2}+\ldots$		(16)
	$\displaystyle\Theta=\Theta_{0}+\operatorname{Re}^{-1/2}\Theta_{1}+\operatorname{Re}^{-1}\Theta_{2}+\ldots$

Substituting (16) into (13) - (15), we get the following three sets of equations:
first order approximation:

\begin{array}[]{r}F_{0}^{\prime\prime\prime}+F_{0}F_{0}^{\prime\prime}-F_{0}^{\prime 2}+1=0\\ \Theta_{0}^{\prime\prime}+\operatorname{Pr}F_{0}\Theta_{0}^{\prime}=0\\ F_{0}(0)=0,F_{0}^{\prime}(0)=0,F_{0}^{\prime}(\infty)=1\\ \Theta_{0}(0)=1,\Theta_{0}(\infty)=0(\mathrm{CWT})\\ \Theta_{0}^{\prime}(0)=-1,\Theta_{0}(\infty)=0(\mathrm{CWHF})\end{array}

second order approximation:

\begin{array}[]{r}F_{1}^{\prime\prime\prime}+F_{0}F_{1}^{\prime\prime}-2F_{0}^{\prime}F_{1}^{\prime}+F_{0}^{\prime\prime}F_{1}+F_{0}^{\prime\prime}+\xi F_{0}^{\prime\prime\prime}=0\\ \Theta_{1}^{\prime\prime}+\operatorname{Pr}\left(F_{0}\Theta_{1}^{\prime}+F_{1}\Theta_{0}^{\prime}\right)+\Theta_{0}^{\prime}+\xi\Theta_{0}^{\prime\prime}=0\\ F_{1}(0)=0,F_{1}^{\prime}(0)=0,F_{1}^{\prime}(\infty)=0\\ \Theta_{1}(0)=0,\Theta_{1}(\infty)=0(\mathrm{CWT})\\ \Theta_{1}^{\prime}(0)=0,\Theta_{1}(\infty)=0(\mathrm{CWHF})\end{array}

third order approximation:

\begin{array}[]{r}F_{2}^{\prime\prime\prime}+F_{0}F_{2}^{\prime\prime}-2F_{0}^{\prime}F_{2}^{\prime}+F_{0}^{\prime\prime}F_{2}+F_{1}^{\prime\prime}+F_{1}^{\prime\prime}F_{1}-F_{1}^{\prime 2}+\xi F_{1}^{\prime\prime\prime}=0\\ \Theta_{2}^{\prime\prime}+\operatorname{Pr}\left(F_{0}\Theta_{2}^{\prime}+F_{1}\Theta_{1}^{\prime}+F_{2}\Theta_{0}^{\prime}\right)+\Theta_{1}^{\prime}+\xi\Theta_{1}^{\prime\prime}=0\\ F_{2}(0)=0,F_{2}^{\prime}(0)=0,F_{2}^{\prime}(\infty)=0\\ \Theta_{2}(0)=0,\Theta_{2}(\infty)=0(\mathrm{CWT})\\ \Theta_{2}^{\prime}(0)=0,\Theta_{2}(\infty)=0(\mathrm{CWHF})\end{array}

3. Results and discussions

Equations (7) - (8) subject to boundary conditions (9) were solved numerically for different values of the Prandtl number ( $\operatorname{Pr}=0.01,0.1,1,10,100$ ) and some values of Reynolds number, $Re=0.01,0.1,0.2,1,10,20,50,100$ using RungeKutta method combined with a shooting technique. Some values related to the Nusselt numbers and skin friction are given in Table 1 for $\operatorname{Pr}=7$ . Results reported by Wang
[5] are also included in this table. It is seen that there is a very good agreement between the present results and those reported by Wang [5]. We are, therefore, confident that our results are very accurate. The validity of the results are also illustrated in Figs. 2 to 4.

Figures 5 to 9 show the dimensionless velocity and temperature profiles for different values of the Reynolds and Prandtl numbers. Thus, it is seen that for a fixed value of the Parndtl number, the velocity profiles increase with the increase of the Reynolds number. However, the temperature profiles decrease with increase of the Reynolds number in the both cases of constant wall temperature and constant heat flux from the plate, respectively, see Figs. 5 to 7. Further, Figs. 8 and 9 show that for the both cases of constant wall temperature and constant heat flux from the plate, temperature profiles decreases with the Parndtl number when the Reynolds number is fixed. As expected the thickness of the temperature boundary layer decreases when the Parndtl number increases.

Finally, Figs. 10 and 11 show the variation of the Nusselt number with the Parndtl number in both cases of constant wall temperature and constant heat flux from the surface for a fixed value of the Reynolds number. The increase of the Nusselt number with the Reynolds number is in agreement with the results given in Table 1.

Table 1: CORNELIA REVNIC, TEODOR GROŞAN, AND IOAN POP

$\eta_{\infty}$	Re	f"(1)	$\theta^{\prime}(1)$	$\theta(1)$
320	0.01	0.313605	-0.320451	3.120599
80	0.1	0.615487	-0.615504	1.624684
35	0.2	0.786053	- 0.780247	1.281645
		0.78605*
11	1	1.484185	- 1.450720	0.689313
		1.484185*
3.5	10	4.162922	- 4.013979	0.249129
		4.16292*
2	20	5.779734	- 5.560052	0.179855
1.75	50	8.985168	- 8.624974	0.115942
1.5	100	12.596429	-12.077699	0.082797
Wang[5]

Table 1. Values of the skin friction, $f^{\prime\prime}(1)$ , Nusselt numbers, $\left(\theta^{\prime}(1)\right.$ for constant temperature case and $\theta(1)$ for the constant wall heat flux case), and boundary layer thickness, $\eta_{\infty}$ , for Prandtl number, $\operatorname{Pr}=7$ and different values of the Reynolds number, Re.

Acknowledgements

Teodor Grosan’s work was supported from the grant CEEX ET90(Romanian Authority of Education and Research). Cornelia Revnic’s work was supported from the grant UEFISCU Grant PN-II-ID-PCE-2007-1/525 (Romanian Ministry of Education and Research).

HEAT TRANSFER IN AXISYMMETRIC STAGNATION FLOW

References

[1] Hiemenz, K., Die Grenzschichat an einem in den gleichfoermingen Fluessigkeitsstrom eingetauchen geraden Kreiszylinder, Dingelers J., 326(1911), 321-410.
[2] Homann, F., Der Einfluss grosser Zähigkeit bei der Strömung um den Zylinder und um die Kugel, Z. Angew. Math. Mech. (ZAMM), 16(1936), 153-164.
[3] Howarth, L., The boundary layer in three-dimensional flow. Part II: The flow near a stagnation point, Phil. Mag. 42(1951), 1433-1440.
[4] Davey, A., Boundary-layer flow at a saddle point of attachment, J. Fluid Mech., 10(1961), 593-610.
[5] Wang, C.-Y., Axisymmetric flow on a cylinder, Quarterly of Applied Mathematics, 32(1974), 207-213.
[6] Gorla, R.S.R., Heat transfer in an axisymmetric stagnation flow on a cylinder, Appl. Sci. Res., 32(1976), 541-553.
[7] Gorla, R.S.R., Unsteady viscous flow in the vicinity of an axisymmetric stagnation point on a circular cylinder, Int. J. Engng. Sci. 17(1979), 87-93.
[8] Gorla, R.S.R., The final approach to steady state in an axisymmetric stagnation flow following a change in free stream velocity, Appl. Sci. Res., 40(1983), 247-251.

"Tiberiu Popoviciu" Institute of Numerical Analysis,
P.O. Box.68-1, Cluj-Napoca, Romania
E-mail address: neli@math.ubbcluj.ro
Faculty of Mathematics and Computer Science,
"Babe-Bolyai" University,
Cluj-Napoca, Romania
E-mail address: tgrosan@math.ubbcluj.ro
Faculty of Mathematics and Computer Science,
"Babe-Bolyai" University,
Cluj-Napoca, Romania
E-mail address: pop.ioan@yahoo.co.uk

Heat transfer in axisymmetric stagnation flow on a thin cylinder

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HEAT TRANSFER IN AXISYMMETRIC STAGNATION FLOW ON A THIN CYLINDER

Abstract

1. Introduction

2. Basic equations

Navier Stokes

3. Results and discussions

Acknowledgements

HEAT TRANSFER IN AXISYMMETRIC STAGNATION FLOW

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