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Fractional-Diffusion Solutions for Transient Local Temperature and Heat Flux V. V. Kulish Assoc. Mem. ASME, Assistant Professor, School of Mechanical and Production Engineering, Nanyang Technological University, Singapore 639798

J. L. Lage Mem. ASME, Associate Professor, Mechanical Engineering Department, Southern Methodist University, Dallas, TX 75275-0337 e-mail: [email protected]

Applying properties of the Laplace transform, the transient heat diffusion equation can be transformed into a fractional (extraordinary) differential equation. This equation can then be modified, using the Fourier Law, into a unique expression relating the local value of the time-varying temperature (or heat flux) and the corresponding transient heat flux (or temperature). We demonstrate that the transformation into a fractional equation requires the assumption of unidirectional heat transport through a semiinfinite domain. Even considering this limitation, the transformed equation leads to a very simple relation between local timevarying temperature and heat flux. When applied along the boundary of the domain, the analytical expression determines the local time-variation of surface temperature (or heat flux) without having to solve the diffusion equation within the entire domain. The simplicity of the solution procedure, together with some introductory concepts of fractional derivatives, is highlighted considering some transient heat transfer problems with known analytical solutions. 关S0022-1481共00兲01002-1兴 Keywords: Conduction, Heating, Heat Transfer, Transient, Unsteady

Introduction Transient, particularly periodic, diffusion problems are very common in practicing engineering. Some examples are the cyclic heating of the cylinder surface of internal combustion engines, the diurnal heating and nocturnal cooling of building structures, lakes and water reservoirs by radiation, the periodic 共pulse兲 laser heating of solid surfaces in materials processing, the cyclic heating of laminated steel during pickling, the periodic heating and cooling of vials contained DNA for polimerase-chain-reaction activation, and the heating of electronics, which is also cyclic in most cases. Frequently, these heat transfer processes are diffusiondominated or at least influenced by the initial diffusion-dominated evolution. Obtaining analytic solutions of transient diffusion problems, when possible, can be very complicated 共as demonstrated in ¨ zisik 关3兴, Kakac¸ detail by Carslaw and Jaeger 关1兴, Arpaci 关2兴, O and Yener 关4兴, and Poulikakos 关5兴兲 because of the mathematical intricacies involved in solving the differential equations governing the phenomenon. The method of choice for solving problems involving timeperiodic temperature boundary conditions analytically, for instance, is the method of complex temperature 共关5兴兲. This method, Contributed by the Heat Transfer Division for publication in the JOURNAL OF HEAT TRANSFER. Manuscript received by the Heat Transfer Division, August 27, 1999; revision received, December 9, 1999. Associate Technical Editor: T. Avedisian.

372 Õ Vol. 122, MAY 2000

however, applies only when the boundary condition is expressed as a sine or cosine function of time. Moreover, the solution does not cover the initial transient regime, but only the steady-periodic regime. Analytic solution of the one-dimensional diffusion problem with a continuous, transient, temperature boundary condition f (t), including the initial transient regime, can be obtained using the Duhamel’s theorem 共关5兴兲 T 共 x,t 兲 ⫽

冕

␶ ⫽t

␶ ⫽0

d f 共␶兲 T s 共 x,t⫺ ␶ 兲 d ␶ d␶

(1)

with T s (x,0)⫽0, where T s (x,t) is the solution for the unit step boundary condition applied at t⫽0. Although very powerful, Duhamel’s theorem can lead to mathematically complex problems because of the integral in Eq. 共1兲, making it very difficult to find an analytic solution except for very simple cases. A more general analytic approach for solving diffusion problems with transient boundary condition exists leading to solutions involving the Green functions and kernels. Even for relatively simple boundary conditions the mathematical analysis can be daunting, again because of the integrals involved, as shown by ¨ zisik 关3兴. O When only localized thermal responses are of interest, the thermal engineer has no choice but to first find the solution to the diffusion equation 共using one of the usual analytic methods兲 throughout the entire domain, and then particularize the solution to a specific location. The possibility of determining analytically the temperature 共or heat flux兲 at a particular location within the domain knowing only the local heat flux 共or temperature兲, i.e., without having to solve the diffusion problem within the entire domain, is very advantageous in these circumstances. The fractional-diffusion technique, originally presented by Oldham and Spanier 关6兴, can be used for determining in a simple and elegant way the local temperature or heat flux of a medium with uniform properties undergoing heat diffusion. Our primary objective is to review this technique, and by doing so to demonstrate why the technique is limited to a semi-infinite and unidimensional configuration, complementing the work by Oldham and Spanier 关6兴. The general analytic solutions 共one for local temperature and one for local heat flux兲 resulting from the fractional-diffusion technique are then validated for several heat diffusion problems with known analytic solutions obtained by different methods considering, in particular, the determination of boundary temperature or heat flux. These validations are important also for demonstrating how easy it is to find analytical solutions using the fractionaldiffusion technique as compared to the mathematical difficulties of other methods.

Fractional Diffusion Equation Consider initially a three-dimensional time-dependent diffusion equation, assuming constant and uniform properties

⳵ T 共 x 1 ,x 2 ,x 3 ,t 兲 ⫺ ␣ ⵜ 2 T 共 x 1 ,x 2 ,x 3 ,t 兲 ⫽0, ⳵t

(2)

where T is temperature, t is time, (x 1 ,x 2 ,x 3 ) are three coordinates of an orthogonal system, and ␣ is the thermal diffusivity of the medium. The system is initially at equilibrium, so T(x 1 ,x 2 ,x 3 ,t)⫽T 0 for t⬍0, with T 0 being a constant and uniform value. Implementing the change of variables ( ␰ 1 , ␰ 2 , ␰ 3 ) ⫽ ␣ ⫺1/2(x 1 ,x 2 ,x 3 ,t) and ␪ (x 1 ,x 2 ,x 3 ,t)⫽T(x 1 ,x 2 ,x 3 ,t)⫺T 0 , Eq. 共2兲 becomes

⳵ ␪ 共 ␰ 1 , ␰ 2 , ␰ 3 ,t 兲 ⫺ⵜ 2 ␪ 共 ␰ 1 , ␰ 2 , ␰ 3 ,t 兲 ⫽0 ⳵t

(3)

with the initial condition now written as ␪ ( ␰ 1 , ␰ 2 , ␰ 3 ,0)⫽0. Taking the Laplace transform of Eq. 共3兲, and using the initial condition ␪ ( ␰ 1 , ␰ 2 , ␰ 3 ,0)⫽0,

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s ␪ * 共 ␰ 1 , ␰ 2 , ␰ 3 ,s 兲 ⫺ⵜ 2 ␪ * 共 ␰ 1 , ␰ 2 , ␰ 3 ,s 兲 ⫽0

(4)

where ␪ * ( ␰ 1 , ␰ 2 , ␰ 3 ,s) represents the Laplace transform of ␪ ( ␰ 1 , ␰ 2 , ␰ 3 ,t). Using the separation of variables ␪ * ( ␰ 1 , ␰ 2 , ␰ 3 ,s) ⫽X 1 ( ␰ 1 )X 2 ( ␰ 2 )X 3 ( ␰ 3 ), Eq. 共4兲 is replaced by the system of equations 1 Xj

冋兿

1 hj

冉

⳵ h ph q ⳵ X j ⳵␰ j h j ⳵␰ j

j

冊册

⫽␭ 2j s, p⫽q⫽ j⫽1,2,3

(5)

␭ 21 ⫹␭ 22 ⫹␭ 23 ⫽1.

with the additional condition: In Eq. 共5兲, h j are multipliers 共or scaling functions兲 of the orthogonal system 共e.g., in spherical coordinates: h 1 ⫽1, h 2 ⫽r, h 3 ⫽r sin ␪) 共see 关7兴, p. 34兲. Considering Cartesian coordinates for simplicity (h 1 ⫽h 2 ⫽h 3 ⫽1), the solution of Eq. 共5兲 is ⬁

X j 共 ␰ j ,s 兲 ⫽

兺

m j ⫽⫺⬁

关 A jm j 共 s 兲 e s

1/2␭

jm j ␰ j

⫹B jm j 共 s 兲 e ⫺s

1/2␭

jm j ␰ j

␪ *⫽

m1

m1

冋

兺 兺 兺 兿

m 1 ⫽⫺⬁ m 3 ⫽0 m 2 ⫽0

j⫽1,2,3

共 A jm j e

s 1/2␭

jm j ␰ j

⫹B jm j e

⫺s 1/2␭

jm j ␰ j

册

兲 .

(7)

Now, observe from Eq. 共7兲 that the equality

⳵ ␪ * 共 ␰ ,s 兲 ⫽⫺s 1/2␪ * 共 ␰ ,s 兲 ⳵␰

(8)

holds only when A jm j ⫽0 or B jm j ⫽0 共depending on ␭ jm j being positive or negative, respectively兲 and when ␭ jm j is unity. The requirements on A jm j and B jm j are automatically satisfied when seeking a finite solution valid for a semi-infinite domain along ␰ j . The requirements on ␭ jm j implies in ␭ 1m 1 ⫽1, therefore ␭ 2m 2 ⫽␭ 3m 3 ⫽0, and the problem degenerates to the unidirectional case. Therefore, Eq. 共8兲 is valid only when the problem is unidirectional and the domain is semi-infinite. It will become clear later on that Eq. 共8兲 is a fundamental necessary step along the derivation of the fractional-diffusion equations. As a consequence, the equations are limited to the semi-infinite and unidirectional case. Aside this limitation, the technique is extremely powerful, as will be demonstrated in the following sections. The development of the fractional-diffusion technique continues by inverting Eq. 共8兲, recognizing first that the inverse Laplace transform commutes with the ⳵/⳵␰ operator, i.e., L ⫺1 关 ⳵ ␪ * ( ␰ ,s)/ ⳵␰ 兴 ⫽ ⳵ 兵 L ⫺1 关 ␪ * ( ␰ ,s) 兴 其 / ⳵␰ ⫽ ⳵ 关 ␪ ( ␰ ,t) 兴 / ⳵␰ . It is also necessary to use the property L 关 ⳵ f ␪ ( ␰ ,t)/ ⳵ t f 兴 ⫽s f L 关 ␪ ( ␰ ,t) 兴 ⫽s f ␪ * ( ␰ ,s), valid for a function ␪ ( ␰ ,t) that satisfies ␪ ( ␰ ,0)⫽0, where ⳵ f / ⳵ t f is the fractional derivative operator of order f. Thus, Eq. 共8兲, on restoring the original variables, becomes

␣ 1/2

⳵ T 共 x,t 兲 ⳵ 1/2关 T 共 x,t 兲 ⫺T 0 兴 ⫽⫺ . ⳵x ⳵ t 1/2

(9)

Using the properties 共A2兲 and 共A10兲 listed in the Appendix, Eq. 共9兲 is rewritten as

␣ 1/2

⳵ T 共 x,t 兲 ⳵ 1/2T 共 x,t 兲 1 ⫽⫺ ⫹ T . ⳵x ⳵ t 1/2 共 ␲ t 兲 1/2 0

(10)

Now, recalling the Fourier Law for the heat flux q ⬙ (x,t)⫽ ⫺k ⳵ T(x,t)/ ⳵ x, valid at any point within the domain, and using Eq. 共10兲 to substitute the ⳵ T(x,t)/ ⳵ x term, we obtain q ⬙ 共 x,t 兲 ⫽

冋

册

k ⳵ T 共 x,t 兲 T0 ⫺ . ␣ 1/2 ⳵ t 1/2 共 ␲ t 兲 1/2

Journal of Heat Transfer

1/2

T 共 x,t 兲 ⫽

␣ 1/2 ⳵ ⫺1/2关 q ⬙ 共 x,t 兲兴 ⫹T 0 . k ⳵ t ⫺1/2

(12)

It is important to emphasise that the transformation of the diffusion Eq. 共2兲 into the extraordinary PDE Eq. 共10兲 is restricted only by the assumption of the domain being semi-infinite and the process unidirectional. One can use this transformation in heat and mass transfer for analysing the early regime within a finite domain during which the diffusion process takes place as if the domain were semi-infinite. In the next section Eqs. 共11兲 and 共12兲 are validated by considering problems with known analytic solutions.

兴 (6)

where A jm j and B jm j are arbitrary functions of s. The solution ␪* becomes ⬁

Therefore, the heat flux at any location within the domain 共including along the boundary兲 can be obtained from Eq. 共11兲 by simple semidifferentiating the temperature T in time at that location. Note that for a given q ⬙ (x,t), the corresponding T(x,t) value can be obtained by inverting Eq. 共11兲, i.e., by taking ⳵ ⫺1/2 关Eq. 共11兲兴/⳵ t ⫺1/2. Using properties 共A3兲, 共A5兲, and 共A8兲, the result is

(11)

Validation for Heat Transfer Problems Consider, for instance, the problem of a semi-infinite planar medium with a time-varying temperature condition along the boundary 共the temperature is considered uniform along the boundary, where x⫽0兲. The one-dimensional heat conduction equation in this case is

⳵ T 共 x,t 兲 ⳵ 2 T 共 x,t 兲 ⫺␣ ⫽0. ⳵t ⳵x2

(13)

The initial and boundary conditions are T(x,0)⫽T 0 , T(0,t) ⫽T b (t) and T(⬁,t)⫽0. Using Eq. 共11兲, the heat flux q b⬙ (t) crossing the boundary of the medium is q ⬙ 共 0,t 兲 ⫽q ⬙b 共 t 兲 ⫽

冋

k d 1/2T b 共 t 兲 1 ⫺ T ␣ 1/2 dt 1/2 共 ␲ t 兲 1/2 0

册

(14)

for any time-varying temperature boundary condition T b (t). To gain confidence on the validity of Eq. 共14兲, consider some simple heat transfer cases with known analytic solution. For instance, the constant boundary temperature problem, i.e., T(0,t) ⫽T b . From Eqs. 共14兲 and 共A10兲 q ⬙b ⫽

冉 冊

1 k ␣ 1/2 ␲ t

1/2

共 T b ⫺T 0 兲 ,

(15)

which is exactly the same as the result obtained by solving the diffusion equation analytically for the temperature within the entire domain, using the similarity method and separation of variables, and then obtaining an expression for the boundary heat flux via the Fourier Law 共see Eq. 共4.43兲, p. 150, of Bejan 关8兴兲. Observe that with the fractional approach the same result is obtained in one simple operation, i.e., finding the semiderivative of the constant Tb . Now, to verify the inverse relation, Eq. 共12兲, consider first the problem of a continuous plane source q ⬙ within an infinite domain. The boundary temperature, according to Eq. 共12兲, is T 共 0,t 兲 ⫽T b 共 t 兲 ⫽

␣ 1/2 d ⫺1/2q ⬙ ⫹T 0 . k dt ⫺1/2

(16)

When q ⬙ is a constant, expression 共A13兲 can be used to evaluate the fractional derivative of Eq. 共16兲, and the result is T b共 t 兲 ⫽

冉冊

␣ 1/2 t 2q ⬙ k ␲

1/2

⫹T 0

(17)

which is exactly the result found via the similarity method, reported by Poulikakos 共关5兴, p. 191兲 for this particular problem. The simplicity and clarity 共it does not require finding a similarity variable兲 of the fractional method is again noteworthy. MAY 2000, Vol. 122 Õ 373

Fig. 1 Time evolution of surface heat flux, Eq. „20…, and the imposed boundary condition „dashed line…: † T b „ t …À T 0 ‡Õ T A Äsin„␻t…

Consider now a more complicated class of problems in which the boundary condition is time-dependent. For instance, when the surface temperature is time-dependent and varies as T b (t)⫽T 0 ⫹T A sin(␻t), the boundary heat flux is obtained from Eq. 共11兲 as q ⬙b 共 t 兲 ⫽

再

冎

k d 1/2关 T 0 ⫹T A sin共 ␻ t 兲兴 1 ⫺ T . ␣ 1/2 dt 1/2 ␲ 共 t 兲 1/2 0

(18)

Using, in order, properties 共A2兲 and 共A10兲, listed in the Appendix, and simplifying the result, q ⬙b 共 t 兲 ⫽

k d 1/2关 T A sin共 ␻ t 兲兴 ␣ 1/2 dt 1/2

(19)

Now, using 共A3兲, 共A6兲, and 共A11兲 into Eq. 共19兲, yields q b⬙ 共 t 兲 ⫽

再 冉

冊

k ␲ T ␻ 1/2 sin ␻ t⫹ ⫺2 1/2⌳ ␣ 1/2 A 4

冋冉 冊 册冎 2␻t ␲

Eq. 共20兲 governs the initial unsteady regime. When time is long enough the contribution of the auxiliary Fresnel function becomes negligible because ⌳共␶兲 approaches zero as ␶ increases 共for instance, at t⬃10␲ / ␻ , 兩 max兵⌳关(2␻t/␲)1/2兴 其 兩 ⬍0.001兲, and the heat flux variation reaches a steady-periodic regime, as shown in Fig. 1. The surface heat flux at the steady-periodic regime, for this particular case of boundary condition, can be obtained also by first modifying the diffusion equation to a complex temperature model, solving the differential equation for the complex temperature, extracting the temperature solution from the complex temperature and then using the Fourier’s law to obtain the corresponding heat flux 共see 关5兴, p. 167兲. The final result is

1/2

(20)

where ⌳ is the auxiliary Fresnel function 共see Appendix兲. Observe that the term proportional to the auxiliary Fresnel function ⌳ in

q ⬙b 共 t 兲 ⫽

冉冊

␻ k 1/2 T A ␣ 2

1/2

关 sin共 ␻ t 兲 ⫹cos共 ␻ t 兲兴

(21)

which is the asymptotic result of Eq. 共20兲 for very large time.

Fig. 2 Time evolution of surface temperature, Eq. „23…, and the imposed boundary condition „dashed line…: q ⬙ „ t …Õ q 0⬙ Äsin„␻t…

374 Õ Vol. 122, MAY 2000

Transactions of the ASME

Consider now the boundary condition q ⬙b (t)⫽q ⬙0 sin(␻t) imposed at the boundary (x⫽0) of a semi-infinite medium. Using the fractional Eq. 共12兲 the boundary temperature is directly related to the boundary heat flux via T b共 t 兲 ⫽

␣ 1/2 d ⫺1/2q ⬙b ␣ 1/2 d ⫺1/2关 q ⬙b sin共 ␻ t 兲兴 ⫽ . k dt ⫺1/2 k dt ⫺1/2

T b共 t 兲 ⫽

冋 冉

冊

冉 冊册

␣ ␲ 2␻t ⫹2 1/2⍀ q ⬙ ␻ 1/2 sin ␻ t⫺ k 0 4 ␲

In this section some useful definitions and properties of fractional derivatives are presented. From the several equivalent definitions of fractional derivatives, the most elegant is the RiemannLiouville definition 共关10兴兲, namely

(22)

The final solution is obtained by applying 共A3兲, 共A6兲, and 共A14兲 to Eq. 共22兲: 1/2

Appendix

1/2

.

d f 关 g 共 t 兲兴 1 ⫽ dt f ⌫共 ⫺ f 兲

T 共 x,t 兲 ⫽ 共 ␻ ␣ 兲

q 0⬙

冋冉 冊 册 冋 冉 冊

␻ exp ⫺ k 2␣

1/2

␻ x sin ␻ t⫺ 2␣

1/2

(23)

q 0⬙ k

冉

sin ␻ t⫺

␲ 4

冊

册

␲ x⫺ . 4 (24)

(25)

which is exactly the same as the asymptotic result of Eq. 共23兲, when the auxiliary Fresnel function contribution becomes negligible. Other more complex functions can be considered as boundary conditions and used in Eqs. 共20兲 and 共23兲, following the same steps demonstrated previously. A very general set of fractional derivatives and rules of derivation can be found in Odham and Spanier 关9兴.

Journal of Heat Transfer

1⫹ f

d␶

(A1)

d f 关 u 共 t 兲 ⫹ v共 t 兲兴 d f 关 u 共 t 兲兴 d f 关v共 t 兲兴 ⫽ ⫹ dt f dt f dt f

(A2)

d f 关 Cg 共 t 兲兴 d f g共 t 兲 ⫽C f dt dt f

(A3)

d f 关 tg 共 t 兲兴 d f g共 t 兲 d f ⫺1 g 共 t 兲 ⫽t f f ⫹f dt dt dt f ⫺1

(A4)

dh d f g共 t 兲 d h⫹ f g 共 t 兲 ⫽ h f dt dt dt h⫹ f

(A5)

d f g 共 Ct 兲 d f 关 g 共 Ct 兲兴 f ⫽C dt f d 共 Ct 兲 f

(A6)

d f ␦ 共 t⫺ ␶ 兲 1 ⫽ 共 t⫺ ␶ 兲 ⫺ f ⫺1 , f ⬍0 dt f ⌫共 ⫺ f 兲

(A7)

⌫ 共 n⫹1 兲 n⫺ f d f 关 tn兴 ⫽ t dt f ⌫ 共 n⫹1⫺ f 兲

(A8)

d f关C兴 Ct ⫺ f f ⫽ dt ⌫ 共 1⫺ f 兲

(A9)

where ␦ (t⫺ ␶ ) is the Dirac delta function, defined as ␦ (t⫺ ␶ ) ⫽⬁, if t⫽ ␶ , otherwise, ␦ (t⫺ ␶ )⫽0. In the previous formulas, C is a nonzero constant. Observe that expression 共A5兲 holds for any positive h and f, and for d ⫺ f 关 d f g(t)/dt f 兴 /dt ⫺ f ⫽g(t) if f and/or h are negative 共see Oldham and Spanier, 关9兴, p. 117兲. The semiderivatives 共case of f being ⫾1/2兲 of some common functions are

⳵ 1/2关 C 兴 ⫽C 共 ␲ t 兲 ⫺1/2 ⳵ t 1/2

Summary and Conclusions By attempting to extend to vectorial form the development presented by Oldham and Spanier 关6兴, it is demonstrated why the fractional-diffusion technique is restricted to unidirectional problems within a semi-infinite domain. The two resulting equations, one for local temperature and one for local heat flux, depend on local quantities only 共heat flux and temperature, respectively兲. Therefore, when the temperature 共or heat flux兲 time evolution is known at any point within the domain, the time evolution of the corresponding heat flux 共or temperature兲 at the same point can be found without having to solve the diffusion equation for the entire domain. The resulting fractional-diffusion equations are applied to the boundary and validated considering several problems with known analytic solution. The simplicity involved in obtaining the local system response 共temperature or heat flux兲 to a transient excitation within a semi-infinite diffusion system using the fractional approach is highlighted. Observe that the extension to mass diffusion problems is straightforward by considering the parallel between heat and mass transfer. The same can be said about mass and heat diffusion within a porous medium following the macroscopic diffusion equation for a system in 共thermal兲 equilibrium. In this case care should be taken when replacing the diffusion coefficient ␣ and the thermal conductivity k with the appropriate effective coefficients of the porous medium.

g共 ␶ 兲

0 共 t⫺ ␶ 兲

冉 冊

At the boundary, x⫽0, the solution simplifies to T b 共 t 兲 ⫽ 共 ␣ ␻ 兲 1/2

t

where f is any negative number and ⌫ is the Gamma function. Some of the useful properties derived from Eq. 共A1兲 are

Observe, again, that the term proportional to the auxiliary Fresnel function ⍀ in Eq. 共23兲 governs the initial unsteady regime 共Fig. 2兲. When time permits the contribution of the auxiliary Fresnel function becomes negligible 共for instance, at t ⬃40␲ / ␻ , 兩 max兵⍀关(2␻t/␲)1/2兴 其 兩 ⬍0.04兲, and the temperature variation reaches a steady-periodic regime. The steady-periodic general solution to this problem is 共see 关3兴, p. 114, to verify the mathematical complexity behind this solution兲 1/2

冕

(A10)

冉 冊 冋冉 冊 册 冉 冊 冋冉 冊 册 冉冊 冉 冊 冋冉 冊 册 冉 冊 冋冉 冊 册

d 1/2关 sin共 t 兲兴 ␲ ⫽sin t⫹ ⫺2 1/2⌳ dt 1/2 4

2t ␲

d 1/2关 cos共 t 兲兴 ␲ 1 ⫽ ⫹cos t⫹ ⫺2 1/2⍀ dt 1/2 4 共 ␲ t 兲 1/2 d ⫺1/2关 C 兴 t ⫺1/2 ⫽2C dt ␲

1/2

2t ␲

(A11)

1/2

(A12)

1/2

d ⫺1/2关 sin共 t 兲兴 ␲ ⫽sin t⫺ ⫹2 1/2⍀ dt ⫺1/2 4

2t ␲

1/2

d ⫺1/2关 cos共 t 兲兴 ␲ ⫽cos t⫺ ⫺2 1/2⌳ dt ⫺1/2 4

2t ␲

1/2

(A13) (A14) (A15)

In Eqs. 共A11兲, 共A12兲, 共A14兲, 共A15兲, ⍀ and ⌳ are the auxiliary Fresnel integrals 共functions f and g, respectively, in Abramowitz and Stegun 关11兴 p. 300兲.

References 关1兴 Carslaw, H. S., and Jaeger, J. C., 1959, Conduction of Heat in Solids, 2nd Ed., Oxford University Press, Oxford, UK.

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关2兴 Arpaci, V. S., 1966, Conduction Heat Transfer, Addison-Wesley, Reading, PA. ¨ zisik, M. N., 1980, Heat Conduction, John Wiley and Sons, New York. 关3兴 O 关4兴 Kakac¸, S., and Yener, Y., 1985, Heat Conduction, Hemisphere, Washington, DC. 关5兴 Poulikakos, D., 1994, Conduction Heat Transfer, Prentice-Hall, Englewood Cliffs, NJ. 关6兴 Oldham, K. B., and Spanier, J., 1972, ‘‘A General Solution of the Diffusion Equation for Semiinfinite Geometries,’’ J. Math. Anal. Appl., 39, pp. 655– 669. 关7兴 Cohen, H., 1992, Mathematics for Scientists and Engineers, Prentice-Hall, Englewood Cliffs, NJ. 关8兴 Bejan, A., 1993, Heat Transfer, John Wiley and Sons, New York. 关9兴 Oldham, K. B., and Spanier, J., 1974, The Fractional Calculus, Academic Press, New York. 关10兴 Riesz, M., 1949, ‘‘L’inte´gral de Riemann-Liuoville et le Proble`me de Cauchy,’’ Acta Math., 81, p. 1. 关11兴 Abramowitz, M., and Stegun, I. A., 1964, Handbook of Mathematical Functions, Dover, New York.

Numerical Study of VortexÕFlame Interaction in Actively Forced Confined Non-Premixed Jets K. R. Anderson Lecturer, Mechanical Engineering Department, California State Polytechnic University, Pomona, 3801 West Temple Avenue, Pomona, CA 91768-4062

S. Mahalingam1 Mem. ASME, Associate Professor, Center for Combustion and Environmental Research, Department of Mechanical Engineering, University of Colorado at Boulder, Campus Box 427, Boulder, CO 80309-0427 e-mail: [email protected]

Numerical Simulation 共DNS兲 of a premixed flame/wall interaction, with provisions for heat release and variable density and viscosity, is considered. They found that turbulent flamelets behave differently as they approach walls. The effects of thermal boundary conditions on premixed flame shape and quenching in duct flows are examined by Hackert et al. 关3兴. Roberts et al. 关4兴 investigate premixed flame quenching induced by flame/vortex interactions in an effort to quantify various regimes of turbulent combustion. Nonpremixed flame/wall interaction studies are surprisingly limited. Peters 关5兴 considered the effects of flame stretch on local quenching in nonpremixed flames using perturbation methods. Wichman 关6兴 employed large activation energy asymptotics to derive an expression for the quenching distance in a nonpremixed flame undergoing a one-step exothermic reaction near a cold wall. The study of Katta et al. 关7兴, which considers quenching in methane-air diffusion flames, is one of the few DNS works that have focused on the dynamics of nonpremixed flame quenching. The effect of extremely close proximity walls on the vorticity distribution in reacting jets is the focus of this work. The implications of such near wall confinement on combustion-driven instabilities can be found in engineering applications such as leanpremixed gas turbine engines and compact hazardous waste incinerators 共关8兴兲. In this paper, time-dependent numerical simulation is used to study the combined effects of increased heat release and semi-confinement on the vorticity dynamics in the near-field region of nonpremixed, actively forced, momentum dominated, coflowing jets. The jets are modeled as unsteady, compressible, reacting flows with temperature-dependent viscosity. Because the primary focus of this study is on the effects heat release and confinement on vorticity production, a simple one-step global reaction model governed by temperature-dependent finiterate Arrhenius kinetics is used to represent the chemistry. The jets are subjected to large-amplitude low-frequency perturbations in order to induce vortex formation.

Problem Formulation A schematic representation of the two-dimensional planar computational domain used is shown in Fig. 1. The stoichiometric mixture fraction is used to examine flame surface dynamics. For

Numerical simulations of coplanar reacting jets subjected to near wall confinement have been performed. The primary conclusion is that for a fixed level of heat release, the mechanism of baroclinic vorticity production increases with more severe wall confinement. 关S0022-1481共00兲00602-2兴 Keywords: Combustion, Computational, Heat Transfer, Jets, Vortex

Introduction Active forcing of reacting flows has recently gained much attention as a means for controlling and stabilizing combustion processes in a variety of engineering applications. Tailoring the combustion process by means of pulsing, or otherwise altering the mass flow rate using injectors placed strategically, has recently been adopted as a novel approach to increasing reactor combustion efficiencies. A comprehensive summary regarding the use of active forcing to tailor combustion processes is provided by Coats 关1兴. The literature regarding premixed flame/wall interactions currently dwarfs the amount of information regarding nonpremixed flame/wall interactions. In the study of Poinsot et al. 关2兴 Direct 1 To whom correspondence should be addressed. Contributed by the Heat Transfer Division for publication in the JOURNAL OF HEAT TRANSFER. Manuscript received by the Heat Transfer Division, January 15, 1999; revision received November 9, 1999. Associate Technical Editor: T. Avedisian.

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Fig. 1 Computational domain showing fuel and oxidizer jets, and boundary conditions

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