A Numerical Study on the Flow Mechanism of Performance Improvement of a Wide-Angle Diffuser by Inserting a Short Splitter Vane

Usage of a wide-angle diffuser may result in unfavorable separated flow and a significant diffuser loss. To improve the performance of the diffusers, inserting short splitter vanes is known as a useful method that has been demonstrated experimentally. Regarding the role of the vane in the diffuser flow, Senoo & Nishi (1977) qualitatively explained that the lift force acting on the vane should be a key factor. However, its quantitative verification remains since then. To challenge this issue, numerical simulations of incompressible flow in a wide angle of 28° two-dimensional diffuser with and without a short splitter vane were conducted in the present study. An improvement of pressure-recovery by the vane and oscillatory flows in the diffuser are reasonably reproduced from comparison with the experimental results made by Cochran & Kline (1958). It is also found that the lift force acting on the vane varies periodically in an opposite phase with the detachment point moved back and forth on a diverging wall, since one vane is not sufficient to fully suppress the flow separation that occurred on the wall and the incoming main-flow shifts toward the other diverging wall in the diffuser. Thus, as a role of splitter vane in the diffuser, “the lift force of the vane is a key factor” may be quantitatively verified from the present numerical simulation. Further, it is confirmed by the local loss analysis that the turbulent kinetic energy production observed in mixing layers contributes most of the loss in the diffuser. Consequently, the present numerical technique may be usable to investigate the flow character in a diffuser with splitter vanes at a design stage. Record Type: Published Article Submitted To: LAPSE (Living Archive for Process Systems Engineering) Citation (overall record, always the latest version): LAPSE:2020.0293 Citation (this specific file, latest version): LAPSE:2020.0293-1 Citation (this specific file, this version): LAPSE:2020.0293-1v1 DOI of Published Version: https://doi.org/10.3390/pr8020143 License: Creative Commons Attribution 4.0 International (CC BY 4.0) Powered by TCPDF (www.tcpdf.org)


Introduction
Diffusers are widely used in fluid machines and model test facilities (e.g., draft tubes of turbines [1][2][3], subsonic wind tunnels [4,5], etc. [6]) as a component to convert the dynamic pressure to the static pressure by decelerating the main flow velocity. If the diffuser loss ∆p Loss is required to design the above-mentioned systems, the following relationships derived from the one-dimensional theory for the incompressible steady flow have been used [7][8][9]: Processes 2020, 8 where ζ d , Cp i , Cp, and η d are the diffuser loss coefficient, the ideal and the actual static-pressure recovery coefficients, and the diffuser effectiveness, respectively. And A R denotes the area-ratio of the diffuser exit area A 2 to the inlet A 1 . p 1 , p 2 , q 1 , V 1 , and ρ are the inlet and the exit static pressures [Pa], the dynamic pressure at the diffuser inlet [Pa], inlet velocity [m/s] and fluid density [kg/m 3 ], respectively. It is known that the diffuser performances given by ζ d and Cp primarily depend on Reynolds number, inflow conditions (i.e., velocity distribution, main-flow turbulence), and diffuser geometries (i.e., divergence angle, area-ratio, and diffuser length) [8]. Taking into account those parameters mentioned above, extensive studies have been conducted experimentally to prepare selection charts of diffuser geometries associated with pressure recovery and flow characteristics as a convenient tool to design two-dimensional [9], conical [10] and annular diffusers [11].
According to the selection chart (or performance map), diffusers with the divergence angle around 7 • may have favorable geometry of duct to provide a low loss, a high pressure-recovery and a stable flow. In some cases of application, however, much wider divergence-angles have to be selected due to the limitation of the axial diffuser length for a given area-ratio. Owing to this selection, the adverse pressure gradient becomes steeper along the diffuser wall, and separation of flow may occur in the diffuser. Once the flow-separation (or stall) takes place in a diffuser, unsteadiness, and non-uniformity of flow are apt to appear at the diffuser exit, and the diffuser loss usually increases together with the decrease of pressure recovery. Thus, many investigations have been carried out to develop separation-control methods including boundary layer suction [12], jet injection [13], high-level turbulence at the inlet [14], passive and active vortex generators [15][16][17], installation of splitter vanes [7,8,[18][19][20], etc. [21]. Note that Sun et al. [22] performed a numerical investigation of the flow rectification of a vaned micro diffuser.
Historically, flow in a diffuser near the stall condition had been treated almost experimentally because the prediction of separation point was hard to achieve from Prandtl's boundary layer analysis, where the interaction between the boundary layer and the adjacent free-stream wasn't considered [23,24]. Furthermore, a fundamental issue from the viewpoint of fluid mechanics was presented by the results from a wide-angle diffuser with splitter vanes [25]. That is, why is the separating flow suppressed by inserting short splitter vanes into the inlet region of a wide-angle diffuser? Though Senoo and Nishi [26] qualitatively explained that the lift force acting on the vane should be a key factor, its quantitative verification still remains to be done.
In order to challenge the unresolved issue mentioned above, numerical simulation of incompressible flow in a wide-angle two-dimensional diffuser with and without a short splitter vane is conducted in this study.

Test Diffuser and Pressure Recovery
In the present study, a specific wide-angle diffuser with and without a splitter vane was investigated. As a test straight diffuser for numerical simulation, we reproduced the geometry used by Cochran and Kline [7]. Referring to their experimental results obtained at the throat Reynolds number of 2.4 × 10 5 , the divergence angle of 28 • and the area-ratio of 4.9 were chosen. Thus, the mean velocity at the diffuser throat is 47.85 m/s in the present cases.
The following pressure recovery coefficient C PR is used to evaluate the diffuser performance in the present simulation for direct comparison with the experimental values:  Figure 1 shows the top view of the computational domain, and its depth is 611 mm. As shown in the figure, upstream and downstream channels are added to the domain to take account for the boundary conditions. Specifically, the upstream channel is 1.5 times the length of its width (width of the upstream channel = 8W 1 ), and the downstream-channel [5] is 3 times the length of its width (width of the downstream channel = 16W 1 ), where W 1 denotes the throat width.  Figure 1 shows the top view of the computational domain, and its depth is 611 mm. As shown in the figure, upstream and downstream channels are added to the domain to take account for the boundary conditions. Specifically, the upstream channel is 1.5 times the length of its width (width of the upstream channel = 8W1), and the downstream-channel [5] is 3 times the length of its width (width of the downstream channel = 16W1), where W1 denotes the throat width. Ideal dry air (20 °C) was used as the working fluid. Its uniform velocity at the inlet of the upstream channel and atmospheric pressure at the outlet of the downstream channel were chosen as the boundary conditions. According to the Reynolds number at the diffuser throat mentioned before, the mean velocity was set as 5.96 m/s at the inlet of the upstream channel. The specification method of "Intensity and Viscosity Ratio" was chosen. The turbulence intensity is 5% and the turbulent viscosity ratio is 10. "No-slip" conditions were imposed on all stationary walls. Figure 2 shows the geometrical parameters of the diffuser with and without a short splitter vane (or flat vane), whose major dimensions are listed in Table 1, where it is seen that the numerical test has only one vane length, which is nearly a quarter of the diffuser length. Note that all wall surfaces were smooth following the experiment [7].   Ideal dry air (20 • C) was used as the working fluid. Its uniform velocity at the inlet of the upstream channel and atmospheric pressure at the outlet of the downstream channel were chosen as the boundary conditions. According to the Reynolds number at the diffuser throat mentioned before, the mean velocity was set as 5.96 m/s at the inlet of the upstream channel. The specification method of "Intensity and Viscosity Ratio" was chosen. The turbulence intensity is 5% and the turbulent viscosity ratio is 10. "No-slip" conditions were imposed on all stationary walls. Figure 2 shows the geometrical parameters of the diffuser with and without a short splitter vane (or flat vane), whose major dimensions are listed in Table 1, where it is seen that the numerical test has only one vane length, which is nearly a quarter of the diffuser length. Note that all wall surfaces were smooth following the experiment [7].  Figure 1 shows the top view of the computational domain, and its depth is 611 mm. As shown in the figure, upstream and downstream channels are added to the domain to take account for the boundary conditions. Specifically, the upstream channel is 1.5 times the length of its width (width of the upstream channel = 8W1), and the downstream-channel [5] is 3 times the length of its width (width of the downstream channel = 16W1), where W1 denotes the throat width. Ideal dry air (20 °C) was used as the working fluid. Its uniform velocity at the inlet of the upstream channel and atmospheric pressure at the outlet of the downstream channel were chosen as the boundary conditions. According to the Reynolds number at the diffuser throat mentioned before, the mean velocity was set as 5.96 m/s at the inlet of the upstream channel. The specification method of "Intensity and Viscosity Ratio" was chosen. The turbulence intensity is 5% and the turbulent viscosity ratio is 10. "No-slip" conditions were imposed on all stationary walls. Figure 2 shows the geometrical parameters of the diffuser with and without a short splitter vane (or flat vane), whose major dimensions are listed in Table 1, where it is seen that the numerical test has only one vane length, which is nearly a quarter of the diffuser length. Note that all wall surfaces were smooth following the experiment [7].    To study the boundary layer and the large separated flow in the diffuser before and after inserting the splitter vane, a low-Reynolds-number RANS (Reynolds Averaged Navier-Stokes) turbulence model, i.e., SST k- To study the boundary layer and the large separated flow in the diffuser before and after inserting the splitter vane, a low-Reynolds-number RANS (Reynolds Averaged Navier-Stokes) turbulence model, i.e., SST k-ɷ turbulence model was applied to this double precision simulation, where wall function was not used. It has been shown that SST k-ɷ has good compatibility in simulating flow separations [27,28]. SIMPLEC (Semi-Implicit Method for Pressure-Linked Equations Consistent) algorithm with a second-order spatial discretization and a first-order implicit transient formulation were chosen as the pressure-velocity coupling method.

Numerical Technique
Structured grid-systems for the whole calculation domain were developed by using commercial software ICEM-CFD (Integrated Computer Engineering and Manufacturing code for Computational Fluid Dynamics) in this study. Figure 3 shows the calculation grids. Figure 3a is the local grid refinements near the edge of the diffuser throat and Figure 3b is the top view of the topology of the mesh. To ensure that the first boundary grid layer is located in a viscous sublayer, required by the SST k-ɷ turbulence model, local grid refinements to the boundary layers were applied at the vane and diffuser walls. Figure 4 shows y + distributions on the diverging walls obtained at steady flow simulation of the diffuser without a vane (or unvaned diffuser). "Ideally, while using enhanced wall treatment, the wall y + should be in the order of 1 (at least less than 5) to resolve the viscous sublayer" [29]. In our simulations, it is controlled less than 5. Unsteady numerical simulations also based on RANS method were applied in the present study. Considering the grid-scale and the Courant number requirement, i.e., C = u∆t/∆s < 10 (where u, ∆t, and ∆s are the characteristic flow velocity, the time step size and typical cell size, respectively), t was set as 0.001 s. The maximum number of iterations was set as 20. Simulation convergence was achieved when the residual error was lower than 10 −6 , or the variation of the operational parameter CPR was below 0.01%. For the purpose of measuring the operational parameter CPR, static pressure monitors were set to record the average static pressure of the diffuser inlet and outlet.
turbulence model was applied to this double precision simulation, where wall function was not used. It has been shown that SST k-

Symbol
Nomenclature Value To study the boundary layer and the large separated flow in the diffuser before an inserting the splitter vane, a low-Reynolds-number RANS (Reynolds Averaged Navierturbulence model, i.e., SST k-ɷ turbulence model was applied to this double precision sim where wall function was not used. It has been shown that SST k-ɷ has good compatib simulating flow separations [27,28]. SIMPLEC (Semi-Implicit Method for Pressure-Linked Eq Consistent) algorithm with a second-order spatial discretization and a first-order implicit tr formulation were chosen as the pressure-velocity coupling method.
Structured grid-systems for the whole calculation domain were developed by using com software ICEM-CFD (Integrated Computer Engineering and Manufacturing code for Compu Fluid Dynamics) in this study. Figure 3 shows the calculation grids. Figure 3a is the loc refinements near the edge of the diffuser throat and Figure 3b is the top view of the topolog mesh. To ensure that the first boundary grid layer is located in a viscous sublayer, required SST k-ɷ turbulence model, local grid refinements to the boundary layers were applied at th and diffuser walls. Figure 4 shows y + distributions on the diverging walls obtained at stead simulation of the diffuser without a vane (or unvaned diffuser). "Ideally, while using enhanc treatment, the wall y + should be in the order of 1 (at least less than 5) to resolve the viscous su [29]. In our simulations, it is controlled less than 5. Unsteady numerical simulations also based on RANS method were applied in the presen Considering the grid-scale and the Courant number requirement, i.e., C = u∆t/∆s < 10 (wher and ∆s are the characteristic flow velocity, the time step size and typical cell size, respective was set as 0.001 s. The maximum number of iterations was set as 20. Simulation convergen achieved when the residual error was lower than 10 −6 , or the variation of the operational par CPR was below 0.01%. For the purpose of measuring the operational parameter CPR, static p monitors were set to record the average static pressure of the diffuser inlet and outlet. has good compatibility in simulating flow separations [27,28]. SIMPLEC (Semi-Implicit Method for Pressure-Linked Equations Consistent) algorithm with a second-order spatial discretization and a first-order implicit transient formulation were chosen as the pressure-velocity coupling method.
Structured grid-systems for the whole calculation domain were developed by using commercial software ICEM-CFD (Integrated Computer Engineering and Manufacturing code for Computational Fluid Dynamics) in this study. Figure 3 shows the calculation grids. Figure 3a is the local grid refinements near the edge of the diffuser throat and Figure 3b is the top view of the topology of the mesh. To ensure that the first boundary grid layer is located in a viscous sublayer, required by the SST k-  study the boundary layer and the large separated flow in the diffuser before and after ng the splitter vane, a low-Reynolds-number RANS (Reynolds Averaged Navier-Stokes) ence model, i.e., SST k-ɷ turbulence model was applied to this double precision simulation, wall function was not used. It has been shown that SST k-ɷ has good compatibility in ting flow separations [27,28]. SIMPLEC (Semi-Implicit Method for Pressure-Linked Equations tent) algorithm with a second-order spatial discretization and a first-order implicit transient lation were chosen as the pressure-velocity coupling method. ructured grid-systems for the whole calculation domain were developed by using commercial re ICEM-CFD (Integrated Computer Engineering and Manufacturing code for Computational Dynamics) in this study. Figure 3 shows the calculation grids. Figure 3a is the local grid ents near the edge of the diffuser throat and Figure 3b is the top view of the topology of the To ensure that the first boundary grid layer is located in a viscous sublayer, required by the ɷ turbulence model, local grid refinements to the boundary layers were applied at the vane ffuser walls. Figure 4 shows y + distributions on the diverging walls obtained at steady flow tion of the diffuser without a vane (or unvaned diffuser). "Ideally, while using enhanced wall ent, the wall y + should be in the order of 1 (at least less than 5) to resolve the viscous sublayer" our simulations, it is controlled less than 5. nsteady numerical simulations also based on RANS method were applied in the present study. ering the grid-scale and the Courant number requirement, i.e., C = u∆t/∆s < 10 (where u, ∆t, are the characteristic flow velocity, the time step size and typical cell size, respectively), t t as 0.001 s. The maximum number of iterations was set as 20. Simulation convergence was ed when the residual error was lower than 10 −6 , or the variation of the operational parameter s below 0.01%. For the purpose of measuring the operational parameter CPR, static pressure rs were set to record the average static pressure of the diffuser inlet and outlet.
turbulence model, local grid refinements to the boundary layers were applied at the vane and diffuser walls. Figure 4 shows y + distributions on the diverging walls obtained at steady flow simulation of the diffuser without a vane (or unvaned diffuser). "Ideally, while using enhanced wall treatment, the wall y + should be in the order of 1 (at least less than 5) to resolve the viscous sublayer" [29]. In our simulations, it is controlled less than 5. To study the boundary layer and the large separated flow in the diffuser before and after inserting the splitter vane, a low-Reynolds-number RANS (Reynolds Averaged Navier-Stokes) turbulence model, i.e., SST k-ɷ turbulence model was applied to this double precision simulation, where wall function was not used. It has been shown that SST k-ɷ has good compatibility in simulating flow separations [27,28]. SIMPLEC (Semi-Implicit Method for Pressure-Linked Equations Consistent) algorithm with a second-order spatial discretization and a first-order implicit transient formulation were chosen as the pressure-velocity coupling method.
Structured grid-systems for the whole calculation domain were developed by using commercial software ICEM-CFD (Integrated Computer Engineering and Manufacturing code for Computational Fluid Dynamics) in this study. Figure 3 shows the calculation grids. Figure 3a is the local grid refinements near the edge of the diffuser throat and Figure 3b is the top view of the topology of the mesh. To ensure that the first boundary grid layer is located in a viscous sublayer, required by the SST k-ɷ turbulence model, local grid refinements to the boundary layers were applied at the vane and diffuser walls. Figure 4 shows y + distributions on the diverging walls obtained at steady flow simulation of the diffuser without a vane (or unvaned diffuser). "Ideally, while using enhanced wall treatment, the wall y + should be in the order of 1 (at least less than 5) to resolve the viscous sublayer" [29]. In our simulations, it is controlled less than 5. Unsteady numerical simulations also based on RANS method were applied in the present study. Considering the grid-scale and the Courant number requirement, i.e., C = u∆t/∆s < 10 (where u, ∆t, and ∆s are the characteristic flow velocity, the time step size and typical cell size, respectively), Δt was set as 0.001 s. The maximum number of iterations was set as 20. Simulation convergence was achieved when the residual error was lower than 10 −6 , or the variation of the operational parameter CPR was below 0.01%. For the purpose of measuring the operational parameter CPR, static pressure monitors were set to record the average static pressure of the diffuser inlet and outlet.  As preliminary tests, predicted CPR of steady flow simulations by using five mesh systems for both unvaned and vaned diffusers were carried out to find the plausible number of grids for the computational domain. The corresponding results are shown in Figure 5 and the mesh sensitivity analysis is summarized in Table 2, where the following features are observed: CPR becomes almost constant in the region of the number of grids greater than 4.2 × 10 5 for the unvaned diffuser and 5.9 × 10 5 for the vaned diffuser. In Table 2, relative change in CPR given by the following expression is used to evaluate the mesh sensitivity. Note that i (2~5) is the serial number of the grid. Unsteady numerical simulations also based on RANS method were applied in the present study. Considering the grid-scale and the Courant number requirement, i.e., C = u∆t/∆s < 10 (where u, ∆t, and ∆s are the characteristic flow velocity, the time step size and typical cell size, respectively), ∆t was set as 0.001 s. The maximum number of iterations was set as 20. Simulation convergence was achieved when the residual error was lower than 10 −6 , or the variation of the operational parameter C PR was below 0.01%. For the purpose of measuring the operational parameter C PR , static pressure monitors were set to record the average static pressure of the diffuser inlet and outlet.
As preliminary tests, predicted C PR of steady flow simulations by using five mesh systems for both unvaned and vaned diffusers were carried out to find the plausible number of grids for the computational domain. The corresponding results are shown in Figure 5 and the mesh sensitivity analysis is summarized in Table 2, where the following features are observed: C PR becomes almost constant in the region of the number of grids greater than 4.2 × 10 5 for the unvaned diffuser and 5.9 × 10 5 for the vaned diffuser. In Table 2, relative change in C PR given by the following expression is used to evaluate the mesh sensitivity. Note that i (2~5) is the serial number of the grid.
Therefore, the mesh system with an 8.3 × 10 5 grid for the unvaned diffuser and that with a 9.7 × 10 5 grid for the vaned diffuser were chosen for the present investigations.  As preliminary tests, predicted CPR of steady flow simulations by using five mesh systems for both unvaned and vaned diffusers were carried out to find the plausible number of grids for the computational domain. The corresponding results are shown in Figure 5 and the mesh sensitivity analysis is summarized in Table 2, where the following features are observed: CPR becomes almost constant in the region of the number of grids greater than 4.2 × 10 5 for the unvaned diffuser and 5.9 × 10 5 for the vaned diffuser. In Table 2, relative change in CPR given by the following expression is used to evaluate the mesh sensitivity. Note that i (2~5) is the serial number of the grid.
Therefore, the mesh system with an 8.3 × 10 5 grid for the unvaned diffuser and that with a 9.7 × 10 5 grid for the vaned diffuser were chosen for the present investigations.    Table 3 shows the comparison of pressure recovery between the steady flow simulation and Cochran & Kline measurement [7], where Equation (5) was applied to the pressure and velocity data at the mid-depth of diffuser for calculation of C PR . If the uncertainty of C PR being estimated between 0.012 and 0.092 in the reference [7] is considered, it may be said that C PR is reasonably predicted by the present simulation. The numerical results also show that an increase of about 43% in C PR is achieved after inserting a short splitter vane in the diffuser. Distributions of pressure in the cross-sections of diffuser inlet (or throat) and the exit are displayed as pressure contours in Figure 6, where the dash-dot line shows the mid-depth. It is seen that the flow characteristics in the mid-depth section may be treatable as the flow in a two-dimensional diffuser since the pressure patterns in the transverse direction (i.e., the horizontal direction in the figure) are not much different in the vicinity of mid-depth. Figure 7 shows the velocity field in the unvaned diffuser. It is seen that flow detachment occurs about 27.5 mm downstream of the throat on one diverging wall (i.e., the lower wall in the present case), and the main through-flow shifts toward the other diverging wall (i.e., upper wall), where no separation is observed. Note that the location of the detachment point [30] was determined by using the limiting streamline method. According to the reference [7], flow characteristics of a wide-angle of 28 • of the two-dimensional diffuser are explained as follows: The flow was relatively steady, and the overall separation that was nearly two-dimensional in form extended to within a small distance from the throat on one diverging wall. In fact, these are observed in Figure 7.

Validation of the Simulation from Diffuser Performance
Consequently, the mid-depth section of the diffuser was chosen for further analyses of computational data. Table 3 shows the comparison of pressure recovery between the steady flow simulation and Cochran & Kline measurement [7], where Equation (5) was applied to the pressure and velocity data at the mid-depth of diffuser for calculation of CPR. If the uncertainty of CPR being estimated between 0.012 and 0.092 in the reference [7] is considered, it may be said that CPR is reasonably predicted by the present simulation. The numerical results also show that an increase of about 43% in CPR is achieved after inserting a short splitter vane in the diffuser. Distributions of pressure in the cross-sections of diffuser inlet (or throat) and the exit are displayed as pressure contours in Figure 6, where the dash-dot line shows the mid-depth. It is seen that the flow characteristics in the mid-depth section may be treatable as the flow in a twodimensional diffuser since the pressure patterns in the transverse direction (i.e., the horizontal direction in the figure) are not much different in the vicinity of mid-depth. Figure 7 shows the velocity field in the unvaned diffuser. It is seen that flow detachment occurs about 27.5 mm downstream of the throat on one diverging wall (i.e., the lower wall in the present case), and the main through-flow shifts toward the other diverging wall (i.e., upper wall), where no separation is observed. Note that the location of the detachment point [30] was determined by using the limiting streamline method. According to the reference [7], flow characteristics of a wide-angle of 28° of the two-dimensional diffuser are explained as follows: The flow was relatively steady, and the overall separation that was nearly two-dimensional in form extended to within a small distance from the throat on one diverging wall. In fact, these are observed in Figure 7.
Consequently, the mid-depth section of the diffuser was chosen for further analyses of computational data.

Flow Characteristics
A red curve in Figure 8 shows the variation of instantaneous pressure-recovery CPR with time in the case of the vaned diffuser. Though the pressure-recovery of the diffuser is really improved, flow steadiness is deteriorated. This may be because one splitter vane cannot fully eliminate the stall zone in the present diffuser. As shown in Figure 9, the fundamental frequency of CPR variation is around 6.1Hz from FFT (Fast Fourier Transform) analysis. In reference [7], it is described that pulsating flow was observed due to the installation of a short vane, but there was no reliable information on its frequency and amplitude. Keeping this in mind, we examined the periodic variation of detachmentpoint location, the distance of which is XS measured from the diffuser throat along the wall, and added it in Figure 8. Understandable are those features that oscillations of CPR and XS are almost inphase, and the minimum CPR appears a little bit later than the minimum XS appears. The occurrence of the time lag may be related to the main flow inertia.

Flow Characteristics
A red curve in Figure 8 shows the variation of instantaneous pressure-recovery C PR with time in the case of the vaned diffuser. Though the pressure-recovery of the diffuser is really improved, flow steadiness is deteriorated. This may be because one splitter vane cannot fully eliminate the stall zone in the present diffuser. As shown in Figure 9, the fundamental frequency of C PR variation is around 6.1 Hz from FFT (Fast Fourier Transform) analysis. In reference [7], it is described that pulsating flow was observed due to the installation of a short vane, but there was no reliable information on its frequency and amplitude. Keeping this in mind, we examined the periodic variation of detachment-point location, the distance of which is X S measured from the diffuser throat along the wall, and added it in Figure 8. Understandable are those features that oscillations of C PR and X S are almost in-phase, and the minimum C PR appears a little bit later than the minimum X S appears. The occurrence of the time lag may be related to the main flow inertia.

Flow Characteristics
A red curve in Figure 8 shows the variation of instantaneous pressure-recovery CPR with time in the case of the vaned diffuser. Though the pressure-recovery of the diffuser is really improved, flow steadiness is deteriorated. This may be because one splitter vane cannot fully eliminate the stall zone in the present diffuser. As shown in Figure 9, the fundamental frequency of CPR variation is around 6.1Hz from FFT (Fast Fourier Transform) analysis. In reference [7], it is described that pulsating flow was observed due to the installation of a short vane, but there was no reliable information on its frequency and amplitude. Keeping this in mind, we examined the periodic variation of detachmentpoint location, the distance of which is XS measured from the diffuser throat along the wall, and added it in Figure 8. Understandable are those features that oscillations of CPR and XS are almost inphase, and the minimum CPR appears a little bit later than the minimum XS appears. The occurrence of the time lag may be related to the main flow inertia.

Flow Characteristics
A red curve in Figure 8 shows the variation of instantaneous pressure-recovery CPR with time in the case of the vaned diffuser. Though the pressure-recovery of the diffuser is really improved, flow steadiness is deteriorated. This may be because one splitter vane cannot fully eliminate the stall zone in the present diffuser. As shown in Figure 9, the fundamental frequency of CPR variation is around 6.1Hz from FFT (Fast Fourier Transform) analysis. In reference [7], it is described that pulsating flow was observed due to the installation of a short vane, but there was no reliable information on its frequency and amplitude. Keeping this in mind, we examined the periodic variation of detachmentpoint location, the distance of which is XS measured from the diffuser throat along the wall, and added it in Figure 8. Understandable are those features that oscillations of CPR and XS are almost inphase, and the minimum CPR appears a little bit later than the minimum XS appears. The occurrence of the time lag may be related to the main flow inertia.    Figure 10 shows six snapshots of the velocity field in the vaned diffuser during one time period T (=0.164 s) of flow oscillation. It is seen that the oscillatory flow is primarily related to the behaviors of the detachment point, and the stall zone extended downstream of the diffuser exit. This phenomenon occurs because one short vane cannot fully suppress the flow separation in the diffuser so that incoming main-flow tends toward the upper wall all the time. Due to this flow condition, the angle of incidence may be larger than the stall angle of a flat vane, and large-scale wake flow from the vane is always observed in the main through-flow, but it doesn't directly contact the boundary layer on the upper diverging wall.
Processes 2020, 8,143 8 of 15 Figure 10 shows six snapshots of the velocity field in the vaned diffuser during one time period T (=0.164 s) of flow oscillation. It is seen that the oscillatory flow is primarily related to the behaviors of the detachment point, and the stall zone extended downstream of the diffuser exit. This phenomenon occurs because one short vane cannot fully suppress the flow separation in the diffuser so that incoming main-flow tends toward the upper wall all the time. Due to this flow condition, the angle of incidence may be larger than the stall angle of a flat vane, and large-scale wake flow from the vane is always observed in the main through-flow, but it doesn't directly contact the boundary layer on the upper diverging wall. To see the pressure rise along the diverging wall, the following dimensionless wall-pressure pw* is introduced: where pw and patm denote local wall-pressure and atmospheric pressure. And is the mean velocity at the throat.
By using Equation (6), six wall-pressure distributions along both upper and lower walls are obtained corresponding to the velocity fields given in Figure 11. From the results shown in Figure 11, where the right end of each pw* curve corresponds to the diffuser exit, the following features are observed:  As atmospheric pressure is given at the domain outlet, dimensionless wall-pressure generally decreases toward diffuser inlet from the diffuser exit.  Since the separation of flow is observed on the lower wall and no separation on the upper wall, recovery of wall-pressure along the upper wall is greater than that along the lower wall.  The steep adverse pressure gradient is observed in the region between the throat and location of the vane's leading-edge.  Depending on the location of the detachment point, the shape of pw* curve varies to some extent. To see the pressure rise along the diverging wall, the following dimensionless wall-pressure p w * is introduced: where p w and p atm denote local wall-pressure and atmospheric pressure. And V 1 is the mean velocity at the throat. By using Equation (6), six wall-pressure distributions along both upper and lower walls are obtained corresponding to the velocity fields given in Figure 11. From the results shown in Figure 11, where the right end of each p w * curve corresponds to the diffuser exit, the following features are observed: • As atmospheric pressure is given at the domain outlet, dimensionless wall-pressure generally decreases toward diffuser inlet from the diffuser exit.

•
Since the separation of flow is observed on the lower wall and no separation on the upper wall, recovery of wall-pressure along the upper wall is greater than that along the lower wall.

•
The steep adverse pressure gradient is observed in the region between the throat and location of the vane's leading-edge.

•
Depending on the location of the detachment point, the shape of p w * curve varies to some extent. detachment point is almost suppressed along the lower wall.
It is noted that pw* curves for unvaned diffuser calculated from the steady flow simulation almost coincide with those of unsteady flow simulation, as the internal flow is fairly steady in the 28° twodimensional diffuser having area-ratio of 4.9 [7].  Time-averaged pressure distributions being derived during a cycle of flow oscillation are shown in Figure 12a for the vaned diffuser and Figure 12b for the unvaned diffuser. In Figure 12a, p w * curves calculated from the steady flow simulation are drawn for comparison. The following features are found from the figures:

•
In the vaned diffuser case, time-averaged p w * curves are almost reproduced by the curves calculated from the steady flow simulation.

•
In the case of the unvaned diffuser, the detachment point occurs much closer to the throat than that of the vaned case, and amounts of wall-pressure recovery reduce greatly.

•
As large stall zone occupies in the unvaned diffuser, pressure rise downstream of the detachment point is almost suppressed along the lower wall.

Force on the Vane
The near-wall velocity-distributions on upper and lower walls at the throat section are shown in Figure 13, where Y is the distance from the center axis of the diffuser. The effects of a splitter vane are clearly observed in the boundary layer of the lower wall. That is, the boundary layer thickness decreases, and the main-flow velocity increases by inserting the vane. It is noted that p w * curves for unvaned diffuser calculated from the steady flow simulation almost coincide with those of unsteady flow simulation, as the internal flow is fairly steady in the 28 • two-dimensional diffuser having area-ratio of 4.9 [7].

Force on the Vane
The near-wall velocity-distributions on upper and lower walls at the throat section are shown in Figure 13, where Y is the distance from the center axis of the diffuser. The effects of a splitter vane are clearly observed in the boundary layer of the lower wall. That is, the boundary layer thickness decreases, and the main-flow velocity increases by inserting the vane.  To investigate the force acting on the vane in a two-dimensional diffuser, the following equation is used to calculate the dimensionless lift FL* assuming that the lift force is approximated by the pressure-force normal to the vane.
where Lv denotes the length of splitter vane. Lift force FL directing toward the upper wall is set as positive.
In fact, the positive value of lift force was always calculated from instantaneous pressures on the vane. After processing those data by using Equation (7), the variation of dimensionless lift with time is correlated in Figure 14, where fluctuation of detachment-point location XS is also plotted for comparison. It is seen that the dimensionless lift oscillates between FL* = 0.15 and FL* = 0.25, having the same fundamental frequency as the pressure recovery and the location of the detachment point. Unlike the pressure-recovery (see Figure 8), the anti-phase character is observed between the location of detachment point XS and the dimensionless lift FL*. The smaller XS is, the larger FL* becomes, and the larger XS is, the smaller FL* becomes. If the lift force acts on a vane in the flow, the reaction force of the vane pushes the flow. Thus, a so-called self-controlling mechanism is recognized in Figure 14, though the flow separation on the lower wall cannot fully be suppressed in the test case. This To investigate the force acting on the vane in a two-dimensional diffuser, the following equation is used to calculate the dimensionless lift F L * assuming that the lift force is approximated by the pressure-force normal to the vane.
where L v denotes the length of splitter vane. Lift force F L directing toward the upper wall is set as positive.
In fact, the positive value of lift force was always calculated from instantaneous pressures on the vane. After processing those data by using Equation (7), the variation of dimensionless lift with time is correlated in Figure 14, where fluctuation of detachment-point location X S is also plotted for comparison. It is seen that the dimensionless lift oscillates between F L * = 0.15 and F L * = 0.25, having the same fundamental frequency as the pressure recovery and the location of the detachment point. Unlike the pressure-recovery (see Figure 8), the anti-phase character is observed between the location of detachment point X S and the dimensionless lift F L *. The smaller X S is, the larger F L * becomes, and the larger X S is, the smaller F L * becomes. If the lift force acts on a vane in the flow, the reaction force of the vane pushes the flow. Thus, a so-called self-controlling mechanism is recognized in Figure 14, though the flow separation on the lower wall cannot fully be suppressed in the test case. This numerical result may quantitatively verify the qualitative explanation for the role of a splitter vane made by Senoo & Nishi [26].

Analysis of Unsteady Local Loss
A local energy analysis method from reference [31] was used to investigate the relationship between the flow patterns and the loss in diffusers [32]. The loss in the diffuser PL is expressed as follows: where μ is the kinematic viscosity and . If a denotes a physical quantity, a and ′ a show the time-averaged value and the fluctuation respectively. Among the five terms, Term I represents the variation of the kinetic energy of the mean flow. Term II and Term III correspond to the diffusion of mean kinetic energy acting by Reynolds stress and viscous stress. Term IV is the turbulent kinetic energy production. And Term V contributes to the loss by viscous dissipation of the mean kinetic energy. As SST k-ɷ turbulence model (RANS) was used in this study, Reynolds stress i j u u ρ ′ ′ was calculated as follows: where μ t , k and δ ij are the eddy viscosity, the turbulent kinetic energy, and the Kronecker delta respectively. The local loss analysis method was applied to see the contribution of each term to the loss in the diffusers. Figure 15 illustrates the terms of loss in unvaned and vaned diffusers separately (Note: λ is the loss factor that is the ratio of each term to the total loss of the mid-depth section in the unvaned diffuser). As the total loss in the vaned diffuser is around 65% of the vaned diffuser loss, the internal flow may be improved by inserting the splitter vane. The results also show that the turbulent kinetic energy production is predominant and it is the primary component of the loss for both diffusers.

Analysis of Unsteady Local Loss
A local energy analysis method from reference [31] was used to investigate the relationship between the flow patterns and the loss in diffusers [32]. The loss in the diffuser P L is expressed as follows:

Analysis of Unsteady Local Loss
A local energy analysis method from reference [31] was used to investigate the relationship between the flow patterns and the loss in diffusers [32]. The loss in the diffuser PL is expressed as follows: where μ is the kinematic viscosity and . If a denotes a physical quantity, a and  a show the time-averaged value and the fluctuation respectively. Among the five terms, Term I represents the variation of the kinetic energy of the mean flow. Term II and Term III correspond to the diffusion of mean kinetic energy acting by Reynolds stress and viscous stress. Term IV is the turbulent kinetic energy production. And Term V contributes to the loss by viscous dissipation of the mean kinetic energy. As SST k-ɷ turbulence model (RANS) was used in this study, Reynolds stress i j u u    was calculated as follows: 1 0 0 where  t , k and  ij are the eddy viscosity, the turbulent kinetic energy, and the Kronecker delta respectively. The local loss analysis method was applied to see the contribution of each term to the loss in the diffusers. Figure 15 illustrates the terms of loss in unvaned and vaned diffusers separately (Note: λ is the loss factor that is the ratio of each term to the total loss of the mid-depth section in the unvaned diffuser). As the total loss in the vaned diffuser is around 65% of the vaned diffuser loss, the internal

Analysis of Unsteady Local Loss
A local energy analysis method from reference [31] was used to investigate the relations between the flow patterns and the loss in diffusers [32]. The loss in the diffuser PL is expressed follows: where μ is the kinematic viscosity and . If a denotes a physical quantity, a and show the time-averaged value and the fluctuation respectively. Among the five terms, Ter represents the variation of the kinetic energy of the mean flow. Term II and Term III correspond to diffusion of mean kinetic energy acting by Reynolds stress and viscous stress. Term IV is the turbu kinetic energy production. And Term V contributes to the loss by viscous dissipation of the m kinetic energy. As SST k-ɷ turbulence model (RANS) was used in this study, Reynolds stress i j u u    was calcula as follows: where  t , k and  ij are the eddy viscosity, the turbulent kinetic energy, and the Kronecker d respectively.
The local loss analysis method was applied to see the contribution of each term to the loss in diffusers. Figure 15 illustrates the terms of loss in unvaned and vaned diffusers separately (Not is the loss factor that is the ratio of each term to the total loss of the mid-depth section in the unva diffuser). As the total loss in the vaned diffuser is around 65% of the vaned diffuser loss, the inte

Analysis of Unsteady Local Loss
A local energy analysis method from reference [31] was used to investig between the flow patterns and the loss in diffusers [32]. The loss in the diffuse follows: where μ is the kinematic viscosity and . If a denotes a physical q show the time-averaged value and the fluctuation respectively. Among the represents the variation of the kinetic energy of the mean flow. Term II and Term I diffusion of mean kinetic energy acting by Reynolds stress and viscous stress. Term kinetic energy production. And Term V contributes to the loss by viscous dissi kinetic energy. As SST k-ɷ turbulence model (RANS) was used in this study, Reynolds stress u  as follows: where  t , k and  ij are the eddy viscosity, the turbulent kinetic energy, and respectively. The local loss analysis method was applied to see the contribution of each te diffusers. Figure 15 illustrates the terms of loss in unvaned and vaned diffusers is the loss factor that is the ratio of each term to the total loss of the mid-depth sec diffuser). As the total loss in the vaned diffuser is around 65% of the vaned diffus

Analysis of Unsteady Local Loss
A local energy analysis method from reference [31] was use between the flow patterns and the loss in diffusers [32]. The loss follows:

Term I Term II Term III Te
where μ is the kinematic viscosity and show the time-averaged value and the fluctuation respectively represents the variation of the kinetic energy of the mean flow. Term diffusion of mean kinetic energy acting by Reynolds stress and visco kinetic energy production. And Term V contributes to the loss by kinetic energy. As SST k-ɷ turbulence model (RANS) was used in this study, Reyn as follows: where  t , k and  ij are the eddy viscosity, the turbulent kinetic respectively. The local loss analysis method was applied to see the contribu diffusers. Figure 15 illustrates the terms of loss in unvaned and va is the loss factor that is the ratio of each term to the total loss of the diffuser). As the total loss in the vaned diffuser is around 65% of th

Analysis of Unsteady Local Loss
A local energy analysis method from refer between the flow patterns and the loss in diffus follows: where μ is the kinematic viscosity and show the time-averaged value and the fluctua represents the variation of the kinetic energy of th diffusion of mean kinetic energy acting by Reyno kinetic energy production. And Term V contribu kinetic energy. As SST k-ɷ turbulence model (RANS) was used i as follows: where  t , k and  ij are the eddy viscosity, the respectively. The local loss analysis method was applied t diffusers. Figure 15 illustrates the terms of loss i is the loss factor that is the ratio of each term to th diffuser). As the total loss in the vaned diffuser is where µ is the kinematic viscosity and D ij = ∂u i ∂x j + ∂u j ∂x i . If a denotes a physical quantity, a and a show the time-averaged value and the fluctuation respectively. Among the five terms, Term I represents the variation of the kinetic energy of the mean flow. Term II and Term III correspond to the diffusion of mean kinetic energy acting by Reynolds stress and viscous stress. Term IV is the turbulent kinetic energy production. And Term V contributes to the loss by viscous dissipation of the mean kinetic energy. As SST k-  To study the boundary layer and the large separated flow in the diffuser before and after inserting the splitter vane, a low-Reynolds-number RANS (Reynolds Averaged Navier-Stokes) turbulence model, i.e., SST k-ɷ turbulence model was applied to this double precision simulation, where wall function was not used. It has been shown that SST k-ɷ has good compatibility in simulating flow separations [27,28]. SIMPLEC (Semi-Implicit Method for Pressure-Linked Equations Consistent) algorithm with a second-order spatial discretization and a first-order implicit transient formulation were chosen as the pressure-velocity coupling method.
Structured grid-systems for the whole calculation domain were developed by using commercial software ICEM-CFD (Integrated Computer Engineering and Manufacturing code for Computational Fluid Dynamics) in this study. Figure 3 shows the calculation grids. Figure 3a is the local grid refinements near the edge of the diffuser throat and Figure 3b is the top view of the topology of the mesh. To ensure that the first boundary grid layer is located in a viscous sublayer, required by the SST k-ɷ turbulence model, local grid refinements to the boundary layers were applied at the vane and diffuser walls. Figure 4 shows y + distributions on the diverging walls obtained at steady flow simulation of the diffuser without a vane (or unvaned diffuser). "Ideally, while using enhanced wall treatment, the wall y + should be in the order of 1 (at least less than 5) to resolve the viscous sublayer" [29]. In our simulations, it is controlled less than 5.
turbulence model (RANS) was used in this study, Reynolds stress ρu i u j was calculated as follows: where µ t, k and δ ij are the eddy viscosity, the turbulent kinetic energy, and the Kronecker delta respectively. The local loss analysis method was applied to see the contribution of each term to the loss in the diffusers. Figure 15 illustrates the terms of loss in unvaned and vaned diffusers separately (Note: λ is the loss factor that is the ratio of each term to the total loss of the mid-depth section in the unvaned diffuser). As the total loss in the vaned diffuser is around 65% of the vaned diffuser loss, the internal flow may be improved by inserting the splitter vane. The results also show that the turbulent kinetic energy production is predominant and it is the primary component of the loss for both diffusers.
Considering that the total loss in the present diffuser is mostly caused by the turbulent kinetic energy production, visualization of its distribution in the vaned diffuser was conducted to see the effect of vane installation. Figure 16 shows such six pictures corresponding to the time used in Figure 10, where the velocity fields are displayed. There are two apparent zones that correspond to the shear layer, where the amount of turbulent kinetic energy production is large. One is observed at the boundary between the main flow and the detached boundary-layer flow (or stall zone) on the lower wall of the diffuser. The other appears in the mixing layer between the main flow and the separating flow from the vane surface (i.e., suction side) faced the upper wall. And it is seen that the highest production occurs in the boundary caused by the leading-edge separation of the vane. Considering that the total loss in the present diffuser is mostly caused by the turbulent kinetic energy production, visualization of its distribution in the vaned diffuser was conducted to see the effect of vane installation. Figure 16 shows such six pictures corresponding to the time used in Figure  10, where the velocity fields are displayed. There are two apparent zones that correspond to the shear layer, where the amount of turbulent kinetic energy production is large. One is observed at the boundary between the main flow and the detached boundary-layer flow (or stall zone) on the lower wall of the diffuser. The other appears in the mixing layer between the main flow and the separating flow from the vane surface (i.e., suction side) faced the upper wall. And it is seen that the highest production occurs in the boundary caused by the leading-edge separation of the vane.

Concluding Remarks
From the numerical simulation of incompressible flow in a wide-angle of 28° two-dimensional diffuser with and without a short splitter vane, the following concluding remarks are drawn:  Considering that the total loss in the present diffuser is mostly caused by the turbulent kinetic energy production, visualization of its distribution in the vaned diffuser was conducted to see the effect of vane installation. Figure 16 shows such six pictures corresponding to the time used in Figure  10, where the velocity fields are displayed. There are two apparent zones that correspond to the shear layer, where the amount of turbulent kinetic energy production is large. One is observed at the boundary between the main flow and the detached boundary-layer flow (or stall zone) on the lower wall of the diffuser. The other appears in the mixing layer between the main flow and the separating flow from the vane surface (i.e., suction side) faced the upper wall. And it is seen that the highest production occurs in the boundary caused by the leading-edge separation of the vane.

Concluding Remarks
From the numerical simulation of incompressible flow in a wide-angle of 28° two-dimensional diffuser with and without a short splitter vane, the following concluding remarks are drawn:

Concluding Remarks
From the numerical simulation of incompressible flow in a wide-angle of 28 • two-dimensional diffuser with and without a short splitter vane, the following concluding remarks are drawn: