1. Introduction
The field of fluid mechanics deals with the flow of fluids and forces us to understand the underlying physics. To study the fluids flow one needs mathematical treatment as well as experimentations. Owing theoretical frame it is well consensus that to study the fluids flow field the simplest mathematical model is Navier-Stokes equations. The both compressible and incompressible flow fields can be studied by coupling the Navier-Stokes equation with stress tensors of concerned fluid models. In this direction, the simpler classical problem of viscous fluid model in two dimensional space was developed by Crane [
1]. The analytical solution was proposed for this problem. Since then many investigations in similar manner were carried to inspect the flow field properties of both Newtonian and non-Newtonian fluid models like Devi et al. [
2] studied flow due to stretched surface in three dimensional frame. An exact solution of Navier-Stokes equations subject to stretched surface was proposed by Smith [
3]. Pop and Na [
4] extended the study by considering time dependent flow field. The mathematical equations were developed by assuming that the viscous fluid flow is attain due to stretching sheet. Later, the developed partial differential equations were converted into ordinary differential equations and power series solution was exercised. The mathematical model for two dimensional stagnation point flow in the presence of heat transfer aspects was proposed by Chaim [
5]. In this attempt it is assumed that the surface was stretched linearly. The numerical solution via shooting method was proposed in this paper. The electrically conducting fluid flow along with convective heat transfer individualities was studied by Vajravelu and Hadjinicolaou [
6]. In this problem they assumed linear stretching of an isothermal sheet. The rest of physical effects includes heat absorption, heat generation and natural convection. The flow field is controlled mathematically in terms of partial differential equations. To narrate flow field characteristic the numerical solution was proposed. Chamkha [
7] mathematically formulated the flow over a non-isothermal stretching sheet manifested with unsteady hydrodynamic and porous medium. The both heat generation and heat absorption effects were carried in a magnetized flow field and numerical solution was reported for this attempt. The mathematical model was constructed for viscoelastic fluid model in the presence of heat transfer aspects by Sarma and Nageswara [
8]. The flow was developed by stretching surface. It is important to note that the power-law surface heat flux was taken along with heat generation, heat absorption and viscous dissipation effects. The asymptotic outcomes were enclosed for temperature. The additional effect of porous medium for the viscoelastic fluid flow along with heat transfer properties was taken by Subhas and Veena [
9]. The heat generation, heat absorption and frictional heating were additional physical effects in this analysis. The developed flow narrating differential system was solved case-wise namely wall heat flux and prescribed surface temperature. Vajravelu and Roper [
10] investigated flow field properties of second grade fluid model in the presence of heat transfer characteristics. The flow was developed via stretching of sheet. The numerical solution was offered for developed fourth order non-linear equations. Yürüsoy and Pakdemirli [
11] found the exact solution for boundary layer equations against non-Newtonian fluid flow due to stretching surface. The third grade fluid model was used as non-Newtonian fluid model. The solution was proposed by using Lie group analysis. One can find the concluding past and recent mathematical study of flow fields subject to various geometric illustration in references [
12,
13,
14,
15,
16,
17,
18,
19,
20,
21,
22,
23,
24,
25,
26,
27,
28,
29,
30,
31,
32]. Over the last few decades the interpretation of the dynamics of lid driven enclosure is topic of great interest for the researchers. In this direction, the numerical solution is the mainstays. One can assessed the motivation towards study of flow fields in cavities in References. [
33,
34,
35,
36,
37,
38,
39]. Owning such motivation the novelty of our problem includes:
Staggered cavity
Newtonian fluid model
Upper wall is moving (Case-I)
Both upper and lower walls are moving parallel (Case-II)
Both upper and lower walls are moving antiparallel (Case-III)
Kinetic energy evaluation
Hybrid meshing
The geometry of problem is shown in
Figure 1. The design of present article is carried in such a way: the limited literature survey of mathematical analysis on flow filed via Navier-Stokes equations is reported in
Section 1. The mathematical formulation for purely viscous fluid flow towards staggered cavity is provided in
Section 2. The directory for numerical scheme is offered in
Section 3. The detail analysis for obtained results for each case is reported in
Section 4. The key assumptions are added as
Section 5.
4. Analysis
The fluid flow narrating differential equations in a staggered enclosure are Equations (10) and (11) along with the boundary constraints provided in Equations (12)–(14) are solved numerically. For better novelty, the staggered cavity as a computational domain is discretized in nine different refinement levels. The lowest refinement level divide the cavity into 332 domain elements and 44 boundary elements.
The geometric illustration is offered in
Figure 2. The level two in which the computational domain consists of 450 domain elements and 56 boundary elements is disclosed in
Figure 3. The refinement level three is consists of 72 boundary elements and 724 domain elements is given in
Figure 4. The four level consists of 108 boundary elements and 1272 domain elements and it is displayed in
Figure 5. The
Figure 6 shows that the level five consists of 1884 domain elements and 136 boundary elements. The six refinement level consists of 2950 domain elements and 172 boundary elements as illustrated in
Figure 7. The level seven consists of 7248 domain elements and 340 boundary elements as shown in
Figure 8. The
Figure 9 depicts the level eight which consists of 18,664 domain elements and 652 boundary elements. Lastly, the highest level is the extremely fine which divides the cavity into 24,014 domain elements and 652 boundary elements is shown in
Figure 10. The primitive variables namely, the velocity and pressure are evaluated at level-9 for all cases. The first case includes the motion of top wall with velocity
and rest of walls are taken zero. The key to the graphs for this case are as follows:
Figure 11 is velocity distribution at Re = 50. The velocity plot at Re = 100, Re = 400 and Re = 1000 are offered in
Figure 12,
Figure 13 and
Figure 14 respectively. The pressure distribution at Re = 50, Re = 100, Re = 400 and Re = 1000 are provided in
Figure 15,
Figure 16,
Figure 17 and
Figure 18 respectively. For better insight the line graph study is executed to examine the
u and
v velocity profiles against variation in µ = 0.02 (Re = 25), µ = 0.01 (Re = 50), µ = 0.0025 (Re = 200), and µ = 0.001 (Re = 500). Such output is offered in
Figure 19 and
Figure 20. In detail, the velocity distribution in staggered cavity is examine at Re = 50 and the outcome is displayed in
Figure 11. It is noticed that the streamlines near the top wall has maximum visibility and there are two secondary vortices appears in the left and right corners of the top wall. There is only one primary vortex appear at the center of upper region. Further, for Re = 100 there is a trifling change in the streamlines of the secondary vortices, see
Figure 12. For Re = 400, the secondary vortex in right corner has prominent as compare to the secondary vortex in the left corner see
Figure 13. The velocity distribution is examined at Re = 1000 and the one primary vortex is observed. The
Figure 14 is plotted in this direction. One can note that the increase in Reynolds number cause prominence of secondary vortices. The corresponding pressure evaluation is recorded for Re = 50, Re = 100, Re = 400 and Re = 1000. In detail,
Figure 15 is pressure distribution at Re = 50. It is noticed that the pressure seems maximum at corners of top wall. The
Figure 16,
Figure 17 and
Figure 18 are pressure plots for Re = 100, Re = 400 and Re = 1000 respectively. It is noticed that the higher values of Reynolds number cause increase in pressure at top corners of staggered cavity.
The
Figure 19 is
u-velocity line graph at different values of µ = 0.02 (Re = 25), µ = 0.01 (Re = 50), µ = 0.0025 (Re = 200), and µ = 0.001 (Re = 500). The significant variation in the u-velocity is observed. Similar is the case for the
v-velocity line graph see
Figure 20. In case-II we assumed that the both top and bottom walls are moving in parallel with velocity
and left and right walls are taken at rest. The velocity distribution is evaluated at level-9 for better accuracy.
Figure 21 depicts the velocity distribution at Re = 50. Once can see that the uniform trends are observed in both upper and lower region of staggered cavity. The total of four secondary vortices in the left and right corners of the top and bottom wall and two primary vortices are appeared. The velocity distribution is inspected at Re = 100. The
Figure 22 is evident in this direction. One can see that the secondary vortices becomes more visible. The velocity distribution at Re = 400 and Re = 1000 is examined and offered with the aid of
Figure 23 and
Figure 24 respectively. One can see that the increase in Reynolds cause significant deformation in primary and secondary vortices. The pressure distribution when upper and lower walls of staggered cavity are moving parallel is examined at various of Reynolds number. The adopted values of Reynolds number are Re = 50, Re = 100, Re = 400 and Re = 1000. To be more specific, the
Figure 25 offers pressure plot at Re = 50. One can see that the pressure seems maximum at corns points of cavity.
Figure 26,
Figure 27 and
Figure 28 are the pressure examination at Re = 100, Re = 400 and Re = 1000 respectively. It is observed that the higher values in Reynolds number results dense pressure at corns of staggered cavity. The
Figure 29 is
u-velocity line graph at different values of µ = 0.02 (Re = 25), µ = 0.01 (Re = 50), µ = 0.0025 (Re = 200), and µ = 0.001 (Re = 500). The significant variation in the
u-velocity is observed while line graph study as
v-velocity towards µ = 0.02 (Re = 25), µ = 0.01 (Re = 50), µ = 0.0025 (Re = 200), and µ = 0.001 (Re = 500) is offered in
Figure 30. It is noticed that the
v-velocity reflects inciting values towards higher values of Reynolds number. The case-III defines the antiparallel motion of the top and bottom wall with same velocity. The finite element simulation is performed by carrying hybrid meshing as a level-9. The primitive variables namely, the velocity and pressure are inspected towards higher values of Reynolds number. The adopted values of Reynolds number are Re = 50, Re = 100, Re = 400 and Re = 1000. In detail, the
Figure 31 shows the velocity distribution at Re = 50 when the upper and lower walls are moving antiparallel. The primary and secondary vortices are formed in both upper and lower region of staggered cavity. The streamlines intersect at center region of cavity. The velocity distribution at Re = 100 is observed and offered in terms of graphical trend see
Figure 32. One can see that the additional loop as a vortex is formed at the central region of staggered cavity.
The velocity distribution when top and bottom walls are moving antiparallel is observed at Re = 400 and Re = 1000. The
Figure 33 and
Figure 34 are plotted in this direction. It can be noticed from
Figure 33 that the central region vortex shrinks by increasing Reynold number that is Re = 100. At Re = 1000, the primary vortices becomes dominant and vortex at central region of staggered shrinks significantly. The pressure plots when both walls namely upper and lower are moving antiparallel are offered at different values of Reynolds number that is Re = 50, Re = 100, Re = 400 and Re = 1000. Particularly,
Figure 35 is pressure plot at Re = 50.
Figure 36 is pressure plot at Re = 100 while the
Figure 37 and
Figure 38 are pressure distribution at Re = 400 and Re = 1000 respectively. Collectively, one can see that the pressure distribution becomes higher in strength against increasing values of Reynolds number. The line graph study of velocity distribution when walls are moving antiparallel is performed for both
u and
v components. In detail, the
Figure 39 is
u-velocity line graph at µ = 0.02 (Re = 25), µ = 0.01 (Re = 50), µ = 0.0025 (Re = 200), and µ = 0.001 (Re = 500) while
Figure 40 offers the
v-velocity line graph study towards µ = 0.02 (Re = 25), µ = 0.01 (Re = 50), µ = 0.0025 (Re = 200), and µ = 0.001 (Re = 500). The trifling sinusoid variation is observed for
v-velocity. The kinetic energy is a very important benchmark quantity for the driven cavity flow problems which shows the momentum scale for the entire flow. It can be defined as
To show the grid convergence, we tabulate the kinetic energy at different refinement levels at a fixed Re = 1000 shown in the
Table 1. We can see that grid convergence is achieved for the kinetic energy at level 9. The variation in kinetic energy is recorded for case-I, case-II and case-III towards Reynolds number (by varying viscosity
µ and fixing
L = 1,
U = 0.5).
Table 2,
Table 3 and
Table 4 are plotted in this regard. In detail,
Table 2 provides the value of kinetic energy when only the upper wall of staggered cavity is moving. The kinetic energy values when both the upper and lower walls are moving parallel are reported in
Table 3. The values of kinetic energy in
Table 4 are recorded when both upper and lower walls of staggered cavity are moving antiparallel.