A Numerical Study on the Oscillating Flow Induced by an Acoustic Field around Coal Particles

In order to investigate the acoustically driven oscillating flow around coal particles in the power plant boiler, the two-dimensional, unsteady mass and momentum conservation equations for laminar flow in spherical coordinates are developed numerically. The velocity field, axial pressure gradient, shear stress, and flow separation angle on the particle surface are carefully analyzed with different values of acoustic Reynolds number and Strouhal number. The minimum frequency required for flow separation is also investigated with different SPL (sound pressure level). The axial pressure gradient, shear stress, and separation angle on the surface are proportional to the magnitude of the oscillating flow velocity at low frequency (∼50Hz). However, those physical quantities have different values at high frequency (∼5000Hz), due to the combined effect of curvature and the flow acceleration.


Introduction
Over the years, considerable researches have been carried out on oscillating flows over bodies of various shapes because of the related engineering application [1].The rates of heat and mass transfer from the sphere are enhanced by the oscillation of the surrounding fluid.This phenomenon has good potential in some areas, such as the pulse combustion [2], food refrigeration [3], and heat exchanger [4,5].
The theoretical study of oscillating flow over sphere dates back to the work of Bassett [6].An experiment had been conducted by Odar and Hamilton [7] on oscillating flow over a sphere in attempt to modify Bassett's solution.Mei [8] proposed a general dynamic equation including the quasisteady drag, history force, and added-mass force in the time domain for particle motions at finite Reynolds number (order of several hundred).It was found that the form of history force obtained performed consistently better than that of Odar and Hamilton.Chang and Maxey [9] investigated the oscillating flow over a sphere with the frequency up to 10 Hz and the Reynolds numbers up to 16.7, by using the numerical calculation method.They found that, at very low Reynolds number, separation took place during the deceleration period and there was no separation during the acceleration period.
Then, Alassar [10] extended the Reynolds number to 200 with the series truncation method and had careful analyses on the separation angle and the wake length.Alassar [10] also investigated the phenomenon of second motion (acoustic streaming) created by the oscillating flow over a sphere, by solving the full Navier-Stokes equations.However, all the researches above on oscillating flow over a sphere have one thing in common that the maximum excursion of sphere over one period of the flow is small, compared to the characteristic size of the particles.
When an acoustic field is applied to enhance the combustion and the heat and mass transfer to and from coal particles, the maximum excursion is large compared to the characteristic size of the coal particles.In this case, Pozrikidis [11] investigated the problem of viscous oscillating flow over a particle at low Reynolds numbers (order of ten), by solving the unsteady stokes equation with boundary-integral method.Ha [12] studied the oscillating flow driven by highintensity acoustic field over a spherical particle (the diameter is 100 m) in the air.Some issues were discussed as well, such as the flow structure, axial pressure gradient, shear stress, and flow separation on the particle surface.Oscillating flow driven by a standing wave around a solid particle (the diameter is at the nanoscale) was investigated by Sadhal [13].
In this paper, the two-dimensional, unsteady mass and momentum conservation equations for laminar flow are solved for the oscillating flow induced by an acoustic field around coal particles.The parameters of flue gas in the real power plant boiler are also taken into account.The flow velocity, axial pressure gradient, shear stress, and flow separation on the coal particle surface are analyzed in different Reynolds numbers and Strouhal numbers.Although there is not much about the mechanism of acoustic enhancement of heat and mass transfer from coal particles, this study highlights its good application prospect in the power plant boiler.

Governing Equation and Physical
Model.The general differential equations can be written as [14] where   ,   are the radial velocity and tangential velocity, respectively.  is the source term and its expression in the radial is different from that in tangential directions, as shown in Table 1.
Figure 1 shows the acoustic field and the particle geometry with some nomenclature.A circumferential symmetry is assumed around an axis which passes through the center of the particle and is parallel to the flow direction.

Boundary Conditions. The governing equation (1) has the following initial and boundary conditions.
Initial Conditions (at  = 0).Consider  =  0 . ( Boundary Conditions (When  > 0).Consider the following: at  = 0 and  (symmetry conditions): and as  → ∞,   = − 1 cos (2) sin In (4), the subscript  in   represents the particle surface.As the boundary conditions constrain to no slip and impermeability,   and   at the particle surface are both zero in (4).In a traveling wave acoustic field with the sound pressure   (the unit is N/m 2 ) and the acoustic impedance    0 (  is the flus gas density,  0 is the sound speed in boiler), the peak value of acoustic velocity  1 can be calculated by where   = 10  and  = (  − 94)/20,   represents the sound pressure level, and the unit is dB.
In most of power plant boilers, the furnace is in the state under negative pressure and high temperature.In this paper, the pressure  0 is fixed at the value of 101225 Pa (negative pressure 100 Pa) and temperature  0 at 1200 ∘ C. The thermophysical properties of flue gas are listed in Table 2. Based on the basic theory of sound wave, the sound speed in boiler could be calculated by  0 = √ 0 /  ( is specific heat ratio,  = 1.36); then it can be known that  0 ≈ 757.37 m/s in boiler.

Meshing and Convergence Criteria.
In the present simulation, the particle diameter is fixed at 100 m which is average size of pulverized coal particles in power plant.The numerical solution domain is chosen as 10 times the particle diameter and is meshed with 360 grids points in the axial direction and 50 in the radial direction.The calculation model is shown in Figure 2.
The conservation equations given by ( 1) are solved, using a finite difference scheme.The numerical solution algorithm is the SIMPLEC.One period is divided evenly into 40 time intervals, so the numerical time step is  = 1/40.During the SIMPLEC iteration procedure, the convergence criteria   are defined as follows: where  old ,  new represent the values of   and   during the previous iteration and that during the present iteration, respectively.  is set to 0.005 in this paper.

The Accuracy of the Calculation Model Validation.
It is well known that steady flow over a solid sphere separates at a Reynolds number (Re) of 20 (based on diameter) and forms a steady axisymmetric wake behind the body.With the increasing Re, this added recirculating zone grows in size but remains essentially steady and axisymmetric up to 212 approximately.When Re is beyond 270, the flow behind the sphere undergoes yet another transition and develops into a fully three-dimensional, time dependent wake with periodic vortex shedding.When Re is between 420 and 800, the flow starts a transition with vortex shedding irregularly [15].In this case by using empirical correlation for the steady Reynolds number range of 10 to 400, the separation angle from the front stagnation point can be expressed as [16]   = 180 − 42.5 [In ( where Re 0 is the steady Reynolds number; Re 0 =  0 /] and  0 is the steady slip velocity.Figure 3 shows the comparison of separation angle obtained from the correlation with the present simulation results.The simulation results are in good accordance with the results from the correlations.

Results and Discussion
The wall shear stress on the particle surface in spherical coordinate system could be calculated from the velocity profile as follows: At the separation point, the wall shear stress   is zero.After the simulation is ran for the case of an oscillating flow around a stationary spherical particle with different acoustic Reynolds numbers (Re 1 , Re 1 =  1 /],  1 is the amplitude of acoustic velocity,  is the particle diameter, and ] is the flus gas kinematic viscosity) and Strouhal numbers (St, St = / 1 ;  is the sound frequency) under the condition that Re 0 = 0.0.The simulation would reach quasisteady state after several cycles.The following discussion concentrates on the results over one cycle right after a quasisteady state, unless it is mentioned otherwise.

The Minimum Frequency Required for Flow Separation in
Different SPL.It is found that flow separation only happens when the frequency is not less than a certain value.As shown in Figure 4, the values of the minimum frequency required for flow separation varies slightly at first and then decreases with the increasing SPL.However, when SPL is over 162 dB, flow separation could be observed in the range of audible frequencies.Therefore, the sound field with SPL = 135 dB ( 1 = 0.875 m/s), 167 dB ( 1 = 34.8m/s) and  = 50 Hz, 5000 Hz has been analyzed in this paper.

Flow Characteristics Induced by Acoustic
Field with SPL = 135 dB.The oscillating flow induced by an acoustic field with SPL = 135 dB around coal particles is discussed in the following paragraph.
Figure 5 shows the velocity vectors in the domain for Re 0 = 0.0 and Re 1 = 0.396 (SPL = 135 dB) corresponding to  = 50 Hz (St = 0.00571) and  = 5000 Hz (St = 0.571).Flow separation (or wake) cannot be observed when = 50 Hz (St = 0.00571), as shown in Figure 5(a).However, if the frequency increases to 5000 Hz (St = 0.571), the flow separation and formation of a wake can be observed even if Re 1 is less than 20, as shown in Figure 5(b).Regarding the unsteady flow, its flow field, separation angle, pressure distribution, and shear stress on the sphere surface are affected by not only the curvature but also the flow acceleration.
The normalized oscillating flow velocity (/ 1 ) and the acceleration ((1/2 1 )(/)) are shown in Figure 6. =  ⋅  represents the dimensionless time.During  = 0.0∼0.25 and  = 0.75∼1.0,/ 1 > 0, which means that the direction is from left to right.The direction is the opposite while  = 0.25∼ 0.75 as / 1 < 0. The axial pressure gradient ((1/)(/)), due to the flow acceleration in the potential flow, is expressed as A new angle  related to the angle  shown in Figure 1 is defined and it is measured from the front stagnation point (streamwise direction).Then during  = 0.0∼0.25 and  = 0.75∼1.0, during  = 0.25∼0.75, By substituting  into (10) with ( 11) and (12), during  = 0∼0.25 and  = 0.75∼1.0, during  = 0.25∼0.75,Table 3 shows a summary on the flow direction, acceleration, and pressure gradient at the infinity due to the imposed acoustic field, where ((1/)(/))  is the axial pressure gradient on the particle surface.APG is the adverse pressure gradient, and FPG is the favorable pressure gradient.
The axial pressure gradient as well as the shear stress on the particle surface for SPL = 135 dB (Re 1 = 0.396) and frequency  = 50 Hz (St = 0.00571), 5000 Hz (St = 0.571) is shown in Figure 7 during the half cycle ( = 0.0∼0.5).
At  = 0.0, the axial pressure gradient is 0.0 for both situations with the frequency of 50 Hz and 5000 Hz since the flow acceleration is 0.0 in the potential flow region, as given by (10) and Table 3.Thus, the curvature and the magnitude of the oscillating flow velocity ( = 0.875 m/s) are the major factors influencing the axial pressure gradient on the sphere surface.The adverse pressure gradient on the sphere surface in the downstream region is not strong enough for flow separation and forming a wake.The value of shear stress on the sphere surface is negative in the streamwise direction for both situations of the frequency of 50 Hz and 5000 Hz, as shown in Figures 7(c) and 7(d).
At  = 0.125, the axial pressure gradient given by ( 10) has positive values resulting in an adverse pressure gradient in the streamwise direction since the flow acceleration is no longer zero.At 50 Hz, the axial pressure gradient given by ( 13) in the potential flow region has the maximum value of 46.6 N/m 3 when  = /2.Similarly, at 5000 Hz, the axial pressure gradient has the maximum value of 4662.7 N/m 3 .At the low frequency of 50 Hz, the axial pressure gradient (varying in the range 0∼46.6 N/m 3 ) is relatively small and has no effect on the axial pressure gradient on the sphere surface.Thus, the axial pressure gradient still depends mainly on curvature and the magnitude of the flow velocity ( = 0.619 m/s).When compared to the flow field at  = 0.0, the flow velocity decreases from 0.875 m/s to 0.619 m/s, resulting in a lower axial pressure gradient and shear stress on the sphere surface which can be observed in Figures 7(a) and 7(c).However, if the frequency increases to 5000 Hz, the adverse pressure gradient (varying in the range 0∼4662.7 N/m 3 ) at infinity is 100 times as large as that at 50 Hz and it is large enough to affect the axial pressure gradient on the surface.As shown in Figure 7(b), the region of the adverse pressure gradient is larger than that at  = 0.0 resulting in flow separation even though the flow velocity decreases.So it can be inferred that the adverse pressure gradient on the surface is not only affected by the magnitude of flow velocity ( = 0.619 m/s) but also by the flow acceleration.
In order to illustrate the effect of flow acceleration on the sphere surface at high frequency, the axial pressure gradient and shear stress on the surface during  = 0.125∼0.25 are investigated, as shown in Figure 8.At 5000 Hz, during  = 0.125∼0.25, the flow velocity  decreases whereas the magnitude of the flow acceleration increases.Therefore, it can be concluded that the effect of curvature and the magnitude of the flow velocity will decrease; meanwhile the effect of flow acceleration on the axial pressure gradient and the shear stress on the sphere surface will increase at 5000 Hz.However, the flow acceleration at 50 Hz is too small to affect the axial pressure gradient on the sphere surface when compared to the value at 5000 Hz.As shown in Figures 8(a) and 8(c), the axial pressure gradient and shear stress on the sphere surface both decrease with decreasing values of the oscillating flow velocity  with no flow separation at 50 Hz.However, the flow acceleration expands the region of the adverse pressure gradient on the sphere surface resulting in a larger wake at 5000 Hz, even though the flow velocity decreases from 0.619 m/s to 0 m/s during  = 0.125∼0.25,as shown in Figure 8(b).It can be also observed that the separation point moves upstream (  = 180 ∘ → 0 ∘ ), as shown in Figure 8(d).
During  = 0.25∼0.5, the axial pressure gradient and shear stress on the sphere surface are mainly affected by the curvature and the magnitude of the flow velocity  at 50 Hz, and the effect of the flow acceleration is negligible.However, at 5000 Hz, the flow acceleration reaches the maximum at  = 0.25, as shown in Figure 7, even though the flow velocity is zero.A flow is formed along the sphere surface and the direction is the same as the oscillating flow during  = 0.25∼ 0.5, as shown in Figure 5(b).This newly formed flow increases the flow velocity compared to the values at 50 Hz.
At  = 0.375, the axial pressure gradient on the sphere surface at 5000 Hz is favorable (negative value) due to the combined effect of the increased flow velocity and the greater flow acceleration with larger values, when compared to the values at 50 Hz.It results in no flow separation, even though the absolute magnitude of the flow velocity is the same as that at  = 0.125 ( = 0.619 m/s), as shown in Figure 7(d).The shear stress at 5000 Hz is larger than that at 50 Hz because of the increased axial pressure gradient, as shown in Figures 7(c  Figure 10 shows the velocity vectors in the simulation domain for Re 0 = 0.0 and Re 1 = 15.75 ( 1 = 34.8m/s, SPL = 167 dB) when the frequency is 50 Hz (St = 0.000144) and 5000 Hz (St = 0.0144).In the potential flow region, during one cycle, the direction of the flow velocity, flow acceleration, and axial pressure gradient for Re 1 = 15.75 are similar to the case when Re 1 = 0.396.However, as the case of Re 1 = 15.75 is much more larger compared to the case of Re 1 = 0.396, even though Re 1 is still less than 20, a wake is formed at both frequency of 50 Hz and 5000 Hz, as shown in Figures 10(a) and 10(b).It can also be observed that the wake size at 5000 Hz is larger than that at 50 Hz.
At 50 Hz, the axial pressure gradient at Re 1 = 15.75 in the potential flow is still too small to affect the axial pressure gradient on the sphere surface, which is six times as large as the value at Re 1 = 0.396.Therefore, the flow field, separation angle, axial pressure gradient, and shear stress on the sphere surface are also affected by the curvature and flow velocity.As for the case at 5000 Hz, flow acceleration becomes one of the most important factors, just similar to the case when Re 1 = 0.396.
Figure 11 shows the axial pressure gradient and shear stress on the particle surface for SPL = 167 dB (Re 1 = 15.75) and  = 50 Hz (St = 0.000144) and 5000 Hz (St = 0.0144) during the half cycle ( = 0.0∼0.5). Figure 12 shows the axial pressure gradient and shear stress for SPL = 167 dB (Re 1 = 15.75) and  = 50 Hz (St = 0.000144) and 5000 Hz (St = 0.0144) during  = 0.125∼0.25.It can be observed in Figures 11(a) and 11(c) that when compared to the results for Re 1 = 0.396 at 50 Hz, the magnitude of the axial pressure gradient and shear stress on the sphere surface during one cycle are much larger.This could explain the fact that the flow separation could always be observed when SPL is larger than 162 dB in the range of audible frequency.The distribution of the axial pressure gradient and shear stress on the sphere surface at Re 1 = 15.75 are similar to those at Re 1 = 0.396.During  = 0.0∼0.25,due to the fact that the flow acceleration reaches its minimum value at  = 0.0 and the maximum value at  = 0.25, an adverse pressure gradient is formed in the potential flow region.Therefore, at  = 5000 Hz, the favorable pressure gradient in the upstream region is decreasing; meanwhile the adverse pressure gradient in the wake is increasing at  = 0.125 and  = 0.25 because of the increasing flow acceleration.The magnitude of the shear stress on the sphere surface is proportional to the magnitude of the axial pressure gradient on the surface, as shown in Figures 11(c) and 11(d).
Similar to the case of Re 1 = 0.396 and  = 5000 Hz, even though the flow velocity  is zero, a flow is formed along with the direction from  = 0 ∘ to 180 ∘ for Re 1 = 15.75 and  = 5000 Hz, as shown in Figure 10(b).The magnitude of its velocity increases with increasing Re 1 .As shown in Figure 11(b), the magnitude of the axial pressure gradient on the sphere surface during  = 0.25∼0.5 increases, due to the combined effect of the newly formed flow and the acceleration of the oscillating flow.The axial pressure gradient and shear stress on the sphere surface during  = 0.5∼0.75 and  = 0.75∼1.0are very similar to those during  = 0.0∼ 0.25 and  = 0.25∼0.5,respectively, except for that they are antisymmetric which have been discussed previously.
Figure 13 shows the distribution of the separation point  along one cycle for Re 1 = 15.75.It can be found that the separation point  at 50 Hz moves downstream from 0 ∘ to 180 ∘ with decreasing magnitude of the oscillating velocity, resulting in the decreasing wake size during  = 0.0∼0.25 and  = 0.5∼0.75.However, the separation point at 5000 Hz moves upstream from 180 ∘ to 0 ∘ resulting in the increasing wake size, due to the increasing effects of flow acceleration (adverse pressure gradient).During  = 0.25∼ 0.5 and  = 0.75∼1.0, the separation point at 50 Hz moves upstream with increased spread.However, at 5000 Hz, the formation of the flow separation is later than that at 50 Hz resulting in the larger values of separation angle  and smaller wake size.

Conclusions
In order to investigate oscillating flow induced by an acoustic field around coal particles in the power plant boiler, the axisymmetric, laminar momentum and mass conservation equations have been solved numerically with different acoustic Reynolds numbers and Strouhal numbers.The following can be concluded from the modelling results.
The axial pressure gradient, shear stress, and flow separation on the sphere surface all depend on the change of the flow velocity  due to body curvature and flow acceleration; the result is in good accordance with Ha's work [17].At low frequency (∼50 Hz), the curvature is the dominant factor, resulting in increasing values of these values with increasing  whereas, at a frequency (∼5000 Hz), the effect of flow acceleration becomes more obvious; there exists the wake with increasing size even though the flow velocity decreases, due to the combined effects of the curvature and the flow acceleration.
It is also found that flow separation would be always observed on the particle surface even though the acoustic Reynolds number is less than 20, only if the sound frequency is larger than a certain value.
When an acoustic field is applied to enhance the combustion for the boiler, flow separation has an important influence on the rate of the heat and mass transfer from the coal particles.Since the rate of heat and mass transfer increases, the combustion efficiency in the power plant boiler would be enhanced.The higher the sound frequency in the range of audio frequency is and the higher the sound pressure level is, the higher the combustion efficiency would be achieved.

Figure 1 :
Figure 1: Schematic diagram showing the geometry and some of the nomenclature used to simulate oscillating flows over a spherical particle in the presence of an acoustic field.

Figure 4 :
Figure 4: The minimum frequency required for flow separation in different SPL.

Table 3 :
Summary of flow direction and axial pressure gradient for an oscillating flow.