Mixed convection with uniform shear flow over horizontal and vertical walls

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Mixed convection with uniform shear flow over horizontal and vertical walls Patrick D. Weidman Department of Mechanical Engineering, University of Colorado, Boulder, CO 80309, USA

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Article history: Received 18 June 2018 Received in revised form 2 September 2018 Accepted 2 October 2018 Available online xxxx

a b s t r a c t A study is made to determine similarity solutions for uniform shear flow along horizontal and vertical heated plates. Only for specified variations of the wall temperature along the plates are these solutions possible. For vertical walls the temperature must decrease as x−1/3 while for horizontal walls it must increase as x1/3 , where x is the streamwise coordinate along each semi-infinite plate. © 2018 Elsevier Masson SAS. All rights reserved.

1. Introduction

2. The vertical wall problem

Previous similarity solutions on forced convection flow past horizontal and vertical plates will now be briefly reviewed. For laminar flow over a vertical heated plate a detailed review, including an exhaustive list of references, may be found in book by Gebhart, et al. [1] as well as in the review papers by Jularia [2] and Chen and Armaly [3]. We will be concerned with the case when the external velocity is directed upward (assisting flow) for which the solutions are unique. These problems have to be solved numerically since similarity solutions generally do not exist. An exception reported in Merkin and Pop [4] is for uniform flow over a vertical plate with temperature varying as x−1 where x is the distance from the leading edge; in this case the velocity boundary layer is described by the Blasius flow and the temperature equation is linearly coupled to the Blasius solution. For uniformly heated horizontal plates a similarity solution exists where again the hydrodynamic flow is governed by the Blasius equation and the temperature is linearly coupled to this solution; see, for example, the book by Incropera and DeWitt [5]. When one applies a convective surface temperature condition for uniform flow over a horizontal plate, Aziz [6] showed that a similarity solution also exists if the heat transfer coefficient in the Robin boundary condition varies as x−1/2 . In this paper we consider conditions for which self-similar solutions exist when the external flow is one of uniform shear over horizontal and vertical flat plates in the presence of uniform gravity. We start with the simpler problem of uniform shear flow over a heated vertical wall in Section 2 and proceed to the slightly more complicated problem of uniform shear flow over a heated horizontal surface in Section 3. A discussion and concluding remarks are given in Section 4.

We use Cartesian coordinates (x∗ , y∗ ) with coordinate velocities (u , v ∗ ). The vertical surface of the plate located at y∗ = 0 and the leading edge is at x∗ = 0. We prescribe the plate temperature and far field velocity as

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∗

T = Tw (x∗ ),

u∗ = β y∗

(2.1)

where β is the shear rate of the vertical external shear flow. The continuity equation is u∗x∗ + vy∗∗ = 0

(2.2)

and the momentum and energy equations simplified by the Boussinesq and boundary-layer approximation takes the form u∗ u∗x∗ + v ∗ u∗y∗ = − u∗ Tx∗ + v ∗ Ty∗ =

1

ρ∞

Px∗∗ + ν u∗y∗ y∗ + α g(T − T∞ )

ν Ty∗ y∗ σ

(2.3a) (2.3b)

where ρ is the fluid density, ν is the kinematic viscosity, α is the coefficient of thermal expansion, σ is the Prandtl number, and P is the dynamic or motion pressure, the hydrostatic pressure having been combined with the body √ √ force. Using the length scale ν/β and the velocity scale βν and introducing a dimensionless temperature T − T∞ = ∆T θ

(2.4)

where ∆T = Tref − T∞ > 0 gives ux + vy = 0

(2.5a)

uux + v uy = −Px + uyy + Gr θ 1 uθx + vθy = θyy

(2.5b)

σ

(2.5c)

https://doi.org/10.1016/j.euromechflu.2018.10.005 0997-7546/© 2018 Elsevier Masson SAS. All rights reserved.

Please cite this article in press as: P.D. Weidman, Mixed convection with uniform shear flow over horizontal and vertical walls, European Journal of Mechanics / B Fluids (2018), https://doi.org/10.1016/j.euromechflu.2018.10.005.

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where we have introduced the Grashof number Gr =

α g ∆T ν 1/2 β 3/2

(2.6)

One could consider transpiration through a porous boundary which would give rise to a pressure gradient along the plate, but that generality will not be considered here. In the present investigation Px = 0. We now look for a similarity solution of the form y (2.7) u(x, y) = φ (x)f ′ (η), θ (x, y) = ψ (x)g(η), η= δ (x) for which the continuity equation gives the plate-normal velocity

v (x, y) = −(δφ )x f (η) + φδx ηf ′ (η).

(2.8)

Inserting the above into the momentum and energy equations yields

δ2 ψ Gr g = 0 φ ( ) δ 2 ψx φ ′ g ′′ + σ δ (δφ )x fg ′ − f g = 0. ψ

f ′′′ + δ (δφ )x ff ′′ − δ 2 φx f ′2 +

(2.9a)

Fig. 1. Variation of vertical wall shear stress parameter f ′′ (0) with Prandtl number for the values Gr = {0.5, 1, 2, 3.5, 5} with the arrow in the direction of increasing Gr.

(2.9b)

In the dimensionless variables the far-field behavior is u → y as y → ∞ which in the similarity variables becomes f ′ (η ) =

δ (x) η φ (x)

(2.10)

which shows that similarity is achieved if δ (x) = Aφ (x). Also for similarity the x-dependencies must disappear and this is done by choosing

δ (δφ )x = C1 ,

δ2 ψ = C3 , φ

δ 2 φx = C2 ,

δ 2 ψx φ = C4 . ψ (2.11)

We seek the simplest forms for the functions dependent on x and this is achieved by choosing A = 1 and C1 = 2/3, C2 = 1/3, C3 = 1, C4 = −1/3 which gives the solutions 1

1

φ (x) = x 3 ,

1

δ (x) = x 3 ,

ψ (x) = x− 3 .

(2.12)

The governing equations now may be written 1 ff ′′ − f ′2 + Gr g = 0 3( 3 ) 2 1 ′ ′′ ′ g +σ fg + f g = 0 3 3 f ′′′ +

2

(2.13a) (2.13b)

which are to be solved with boundary and far-field conditions f (0) = 0,

f ′ (0) = 0,

g(0) = 1,

f ′′ (∞) = 1,

g(∞) = 0.

3. The horizontal wall problem

(2.14)

in which µ is the absolute viscosity of the fluid. Also of interest is the wall heat transfer

√ ⏐ β k∆T ′ ∂ T ⏐⏐ qw = −k ∗ ⏐ =− g (0) ∂ y y∗ =0 ν x2/3

stress parameters f ′′ (0) are presented in Fig. 1 and the wall heat transfer parameters g ′ (0) are displayed in Fig. 2. Note that the wall stress increases monotonically with Gr and that the heat transfer parameter tends to zero as σ → 0. Similarity velocity and temperature profiles f ′ (η) and g(η) for the above listed Grashof numbers at σ = {0.1, 1.0, 10.0} are shown in Figs. 3, 4 and 5, respectively.

(2.13c)

Of particular interest is the wall shear stress given as

⏐ ∂ u∗ ⏐⏐ τw = µ ∗ ⏐ = µβ f ′′ (0) ∂ y y∗ =0

Fig. 2. Variation of vertical wall heat transfer parameter g ′ (0) with Prandtl number for the values Gr = {0.5, 1, 2, 3.5, 5} with the arrow in the direction of increasing Gr.

(2.15)

where k is the fluid thermal conductivity. All numerical calculations reported here are numerically solved using a shooting technique with the ODEINT code of Press et al. [7]. We first carried out a survey of results varying σ at the fixed Grashof numbers Gr = {0.5, 1.0, 2.0, 3.5, 5.0}. The wall shear

Again we use Cartesian coordinates (x∗ , y∗ ) with coordinate velocities (u∗ , v ∗ ), but now horizontal surface of the plate is located at y∗ = 0 and the leading edge is at x∗ = 0. The plate temperature and far field velocity are prescribed as T = Tw (x∗ ),

u∗ = β y∗

(3.1)

where β is the shear rate of the horizontal external shear flow. The continuity equation is u∗x∗ + vy∗∗ = 0

(3.2)

and the momentum and energy equations are simplified by the Boussinesq and boundary-layer approximates to the form u∗ u∗x∗ + v ∗ u∗y∗ = −

1

ρ∞

Px∗∗ + ν u∗y∗ y∗

(3.3a)

Please cite this article in press as: P.D. Weidman, Mixed convection with uniform shear flow over horizontal and vertical walls, European Journal of Mechanics / B Fluids (2018), https://doi.org/10.1016/j.euromechflu.2018.10.005.

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Fig. 3a. Vertical plate velocity profiles for Gr = {0.5, 1, 2, 3.5, 5} at σ = 0.1 with the arrow in the direction of increasing Gr.

Fig. 3b. Vertical plate temperature profiles for Gr = {0.5, 1, 2, 3.5, 5} at σ = 0.1 with the arrow in the direction of increasing Gr.

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Fig. 4b. Vertical plate temperature profiles for Gr = {0.5, 1, 2, 3.5, 5} at σ = 1.0 with the arrow in the direction of increasing Gr.

Fig. 5a. Vertical plate velocity profiles for Gr = {0.5, 1, 2, 3.5, 5} at σ = 10 with the arrow in the direction of increasing Gr.

Fig. 4a. Vertical plate velocity profiles for Gr = {0.5, 1, 2, 3.5, 5} at σ = 1.0 with the arrow in the direction of increasing Gr. Fig. 5b. Vertical plate temperature profiles for Gr = {0.5, 1, 2, 3, 5} at σ = 10 with the arrow in the direction of increasing Gr.

0=−

1

ρ∞

Py∗∗ + g α (T − T∞ )

u∗ Tx∗ + v ∗ Ty∗ =

ν Ty∗ y∗ σ

(3.3b) (3.3c)

where again ρ is the fluid density, ν is the kinematic viscosity, α is the coefficient of thermal expansion, σ is the Prandtl number,

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and P is the dynamic or motion pressure, the hydrostatic pressure having been combined with √ √the body force. Using the length scale ν/β and the velocity scale βν and introducing a dimensionless temperature T − T∞ = ∆T θ

(3.4)

where ∆T = Tref − T∞ > 0 gives ux + vy = 0

(3.5a)

uux + v uy = −Px + uyy

(3.5b)

Py = Gr θ

(3.5c)

uθx + vθy =

1

σ

θyy

(3.5d)

where again the Grashof number Gr =

α g ∆T . ν 1/2 β 3/2

(3.6)

Fig. 6. Variation of horizontal wall shear stress parameter f ′′ (0) with Prandtl number for Gr = {0.5, 1, 2, 3.5, 5} with arrow in the direction of increasing Gr.

We now look for a similarity solution of the form u(x, y) = φ (x)f ′ (η),

θ (x, y) = ψ (x)g(η),

P(x, y) = γ (x)h(η),

η=

y

(3.7)

δ (x)

for which the continuity equation again gives the plate-normal velocity

v (x, y) = −(δφ )x f (η) + φδx ηf ′ (η).

(3.8)

Inserting the above into the momentum and energy equations yields f ′′′ + δ (δφ )x ff ′′ − δ 2 φx f ′2 +

γ δδx ′ δ 2 γx ηh − h=0 φ φ

(3.9a)

ψδ Grg γ ( ) δ 2 ψx φ ′ g ′′ + σ δ (δφ )x fg ′ − f g =0 ψ h′ =

(3.9b) (3.9c)

Similar to the vertical plate problem, the far-field behavior u → y as y → ∞ requires δ (x) = φ (x). Also, for similarity the x-dependencies in Eq. (3.9) must disappear and this is done by choosing

δ (δφ )x =

2 3

,

ψδ = 1, γ

δ 2 φx =

1 3

,

γ δδx 1 = , φ 3

δ 2 γx 2 = , φ 3

1

(3.10)

1

2

δ (x) = x 3 ,

1

γ (x) = x 3 ,

ψ (x) = x 3 . (3.11)

The governing equations are now written as 2 ′′ 1 1 2 ff − f ′2 + ηh′ − h = 0 3 3 3 3 h′ = Gr g ( ) 2 ′ 1 ′ ′′ g +σ fg − f g = 0 3 3 f ′′′ +

(3.12a) (3.12b) (3.12c)

which are to be solved with boundary f (0) = 0,

f ′ (0) = 0,

g(0) = 1,

h′ (0) = Gr

(3.12d)

and far-field conditions f (∞) = 1, ′′

g(∞) = 0,

The wall shear stress is given as

⏐ ∂ u∗ ⏐⏐ = µβ f ′′ (0) τw = µ ∗ ⏐ ∂ y y∗ =0

(3.13)

and the wall heat transfer is

δ 2 φψx 1 = ψ 3

which gives the solutions

φ (x) = x 3 ,

Fig. 7. Variation of horizontal wall heat transfer parameter g ′ (0) with Prandtl number for Gr = {0.5, 1, 2, 3.5, 5} with arrow in the direction of increasing Gr.

√ ⏐ ∂ T ⏐⏐ β qw = −k ∗ ⏐ =− k∆Tg ′ (0) ∂ y y∗ =0 ν

(3.14)

where µ and k are as previously defined. Again a survey of results varying σ at the fixed Grashof numbers Gr = {0.5, 1.0, 2.0, 3.5, 5.0} was carried out. The wall shear stress parameters f ′′ (0) are presented in Fig. 6, the wall heat transfer parameters g ′ (0) are shown in Fig. 7 and the variation of the initial value h(0) required for solution is displayed in Fig. 8. Similar trends in f ′′ (0) and g ′ (0) as for the vertical plate problem are observed. Similarity velocity and temperature profiles f ′ (η) and g(η) for the above listed Grashof numbers at σ = {0.1, 1.0, 10.0} are shown in Figs. 9, 10 and 11, respectively. We note that the trends of the profiles are quite similar to those displayed in Figs. 3, 4 and 5 for the vertical wall problem. 4. Discussion and conclusion

h(∞) = 0.

(3.12e)

To solve the problem one needs to iterate on guesses for f ′′ (0), g ′ (0) and h(0) to obtain the far-field conditions in (3.12e).

Similarity solutions are found for uniform shear flow parallel to horizontal and vertical walls. The existence of these solutions pivots on the required variation of wall temperature along the

Please cite this article in press as: P.D. Weidman, Mixed convection with uniform shear flow over horizontal and vertical walls, European Journal of Mechanics / B Fluids (2018), https://doi.org/10.1016/j.euromechflu.2018.10.005.

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Fig. 8. Variation of h(0) with Prandtl number for Gr = {0.5, 1, 2, 3.5, 5} for a horizontal wall with arrow in the direction of increasing Gr.

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Fig. 10a. Velocity profiles for the horizontal plate for Gr = {0.5, 1, 2, 3.5, 5} at σ = 1.0 with the arrow in the direction of increasing Gr.

Fig. 9a. Horizontal plate velocity profiles for Gr = {0.5, 1, 2, 3.5, 5} at σ = 0.1 with the arrow in the direction of increasing Gr. Fig. 10b. Horizontal plate temperature profiles for Gr = {0.5, 1, 2, 3.5, 5} at σ = 1.0 with the arrow in the direction of increasing Gr.

Fig. 9b. Horizontal plate temperature profiles for Gr = {0.5, 1, 2, 3.5, 5} at σ = 0.1 with the arrow in the direction of increasing Gr.

surfaces: for vertical walls the temperature must decrease as x−1/3 while for horizontal walls it must increase as x1/3 . In both cases the velocity along the wall varies as x1/3 and the boundary layer thickness also varies as x1/3 . An interesting feature is that the wall shear stress τw in both cases in independent of the streamwise coordinate x. However,

Fig. 11a. Velocity profiles for the horizontal plate for Gr = {0.5, 1, 2, 3.5, 5} at σ = 10.0 with the arrow in the direction of increasing Gr.

while the wall heat transfer qw along the horizontal wall is independent of x, it varies as x−2/3 along the vertical wall.

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investigation is linear in the far field, other scenarios are possible, such as the power-law shear flows reported by Weidman, et al. [8]. If solutions exist for either horizontal or vertical walls, they might well lead to a one-parameter family of solutions depending on the exponent of the external power-law shear flow. This remains a problem for future investigation. References

Fig. 11b. Horizontal plate temperature profiles for Gr = {0.5, 1, 2, 3.5, 5} at σ = 10.0 with the arrow in the direction of increasing Gr.

One may think about future situations for which similarity solutions exist for flow along heated horizontal and vertical walls. While it is clear that the variation of velocity in the present

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