Investigation on unsteady characteristics of flow separation in a supercritical carbon dioxide centrifugal compressor

As one of the core components of supercritical carbon dioxide (S-CO2) closed Brayton cycle, the efficiency of S-CO2 centrifugal compressor plays a crucial role. Affected by the special thermophysical properties of S-CO2 fluid, the flow mechanism of S-CO2 centrifugal compressor is different from the centrifugal compressor of conventional fluid, such as air. Previous studies have found that the flow separation within the flow domain of this kind of compressor is easily to occur downstream the pressure surface. Many steady computational fluid dynamics (CFD) studies have been conducted on the S-CO2 centrifugal compressors, but few studies focused on the unsteady evolution of this flow separation. In this paper, the unsteady CFD simulation is carried out in the flow passage of a S-CO2 centrifugal compressor. The solution domain of CFD simulation includes compressor blades, diffuser and volute. The performance and the unsteady flow behavior of S-CO2 centrifugal compressor is obtained. Under low flow rate conditions, the flow separation on the pressure surface of the blade of S-CO2 centrifugal compressor is more likely to occur, causing the decrease of compressor performance. This flow separation has strong unsteady characteristics, which deteriorates several times during a rotation cycle, meanwhile the vortex shedding happens. At the time steps of vortex shedding, the pressure near the trailing edge of the impeller fluctuates greatly, indicating that this unsteady flow separation brings a large flow loss to the S-CO2 centrifugal compressor.


Introduction
Supercritical carbon dioxide (S-CO2) closed Brayton cycle has been seen as one of the most prospective power cycles due to its high efficiency and compact component structure [1].It is expected to be widely used in concentrated solar energy [2,3], nuclear power [4,5], solid oxide fuel cell power system [6,7] and engine waste heat recovery [8,9].As a crucial component, the S-CO2 centrifugal compressor needs to operate with high efficiency and stability to maintain the performance of the S-CO2 Brayton cycle [10,11].
Affected by the variable thermophysical properties of S-CO2 fluid near the critical point, the flow mechanism of S-CO2 centrifugal compressor is different from the centrifugal compressor of fluid with constant properties [12,13].A study group from University of Duisburg-Essen [14] indicated that the real gas effect of S-CO2 fluid can greatly affect the stable operation of S-CO2 centrifugal compressors.During their numerical study [15] , flow separation occurs downstream the impeller domain inside the S-CO2 centrifugal compressor.This flow separation was also noted by Firas et al. [16], who pointed out that it would bring a large flow loss to the S-CO2 centrifugal compressor.According to Bao et al. [17], the flow separation on the blade pressure surface downstream the S-CO2 centrifugal compressor tended to deteriorate further at lower inlet temperatures, while the circumferential inhomogeneity caused by the volute structure will also be exacerbated.The method to weaken this kind of flow distortion so as to improve the performance of the S-CO2 centrifugal compressor was studied by Xia et al. [18] and Shi et al. [19].
Previous studies have shown that this flow separation downstream the impeller domain is an important factor leading to the performance degradation of the S-CO2 centrifugal compressor.However, most of the studies focused on the steady state of this flow separation, and their method to study this flow separation is steady computational fluid dynamics (CFD), while little attention has been paid to the unsteady characteristics of this flow separation downstream the impeller domain of the S-CO2 centrifugal compressor.
In this paper, the unsteady flow characteristics of the flow separation downstream the impeller domain inside a S-CO2 centrifugal compressor is studied.The performance and flow behavior of the S-CO2 centrifugal compressor is obtained by CFD simulation.As the flow is easier to separate on the blade pressure surface when the S-CO2 centrifugal compressor operates at lower flow rates, unsteady CFD simulation is carried out at a low flow rate condition.The unsteady evolution of this kind of separation and the downstream vortex resulted by it is analyzed.

The S-CO2 Centrifugal Compressor
The study object of this paper is a centrifugal compressor with S-CO2 working fluid designed by Tsinghua University.The main geometries and operating conditions of this centrifugal compressor are shown in Table 1.Shrouded impellers are employed during the design process, including 12 blades.The diffuser is designed to be vaneless.Figure 1 shows the compressor geometry studied in this paper.

CFD Method
As one of the CFD solvers friendly to turbomachinery, ANSYS CFX is chosen to carried out the CFD simulation of internal flow of the S-CO2 centrifugal studied in this paper.K-epsilon turbulence model is employed to close the Reynolds-averaged Navier-Stokes (RANS) equations.Boundary conditions include specified values for the inlet mass flow rate, inlet total temperature, and outlet static pressure.Adiabatic non-slip walls characterize all surfaces of the flow domain.Convergence is determined by a global residual criterion set at 10 -8 .As the source of the thermophysical properties of CO2, NIST REFPROP database is used.Based on that database, real gas properties (RGP) file is created as interpolation table.Thermodynamic parameters such as density and enthalpy are obtained as functions of temperature and pressure in the interpolation table based on the equation of state (EOS) of CO2 proposed by Span and Wanger [20].The ranges of the fluid pressure and temperature are set as 1-25MPa and 243-393K, respectively, including the boundaries of pressure and temperature at which the S-CO2 centrifugal compressor operates.In order to strike a balance between computational efficiency and accuracy, this study utilizes interpolation tables featuring 200×200 interpolation points for pressure and temperature in numerical calculations.The interpolation interval for temperature is set as 0.75 K, and 0.12MPa for pressure.
For unsteady numerical calculation, the sliding mesh method is employed to address the rotationalstatic interface between the impeller domain and the diffuser.With the aim of accurate and effective calculation for the unsteady evolution of the flow characteristics, 30 time steps is set for each passage of the impeller domain.The total number of computational time steps within a rotational period is set to 360, corresponding to a time step length of approximately 0.00595ms.During each time step, the impeller of the SCO2 centrifugal compressor is rotated 1 degree.
The calculated domains in this paper include the inlet domain, the impeller domain, the vaneless diffuser and the volute.The volute domain is meshed by unstructured meshes by ANSYS Meshing, while other domains is meshed by structural meshed by ANSYS TurboGrid.In order to get closer to the real flow situation, only the impeller domain is set as the rotating component during the CFD simulation, and the rest domains are set as the stationary components.Eight different mesh configurations with varying quantities are generated.The independence of these grids is verified, and the findings are depicted in Figure 2. The normalized pressure ratio exhibits a variation of less than 0.13% when the grid count surpasses 4×10 7 .Hence, the numerical outcomes are deemed independent as the grid element count reaches 4×10 7 .The subsequent mesh used in further exploration comprises 4071362 nodes, as illustrated in Figure 3.The y+ values for the near-wall first layer grid are maintained between 30 and 120 as illustrated in Figure 4, satisfying the demand of k-epsilon turbulence model.

CFD Validation
In order to affirm the dependability of the numerical approach, the S-CO2 centrifugal compressor designed and tested by UDE[21] in 20000r/min is chosen as the object of validation, as shown in Figure 5.The reason for this selection is that the impellers of this centrifugal compressor are closed impellers, as well as the S-CO2 centrifugal compressor studied in this paper.The method outlined in the previous section was used to generate a mesh containing a total of 2,864,473 nodes.Figure 6 illustrates the performance curve for the UDE S-CO2 centrifugal compressor, comparing experimental data with the results obtained through CFD simulations in this paper.As the mass flow rate decreases, the total pressure ratio trends from the CFD simulations closely align with those from the experiments.However, the CFD results exhibit a slight elevation compared to the experimental data.This discrepancy can be primarily attributed to factors such as

Compressor Aerodynamic Performance
In this study, the aerodynamic performance for the S-CO2 centrifugal compressor is obtained by steady numerical calculation.The flow coefficient ϕ of the S-CO2 centrifugal compressor, defined as Equation ( 1), is changed by varying the inlet mass flow rate.
Where ρin is the inlet density, U2 is the velocity at the exit of the impeller, D2 is the diameter of the exit of the impeller.The steady CFD simulations are carried out under various flow coefficients.
The total pressure ratio of the S-CO2 centrifugal compressor at different flow coefficients is illustrated in Figure 7.The flow coefficient ϕ is normalized by the that at the design point ϕdesign.With the decrease of the flow coefficient ϕ, the total pressure ratio of the S-CO2 compressor increases firstly and then decreases.As the flow coefficient ϕ is lower than that of design point, the rise rate of the total pressure ratio slows down significantly.Finally, the total pressure ratio drops at the lowest flow coefficient point.The observed trend in the total pressure ratio suggests the presence of flow distortion within the S-CO2 centrifugal compressor.This distortion becomes more pronounced and hinders the increase in the total pressure ratio as the compressor operates at lower flow coefficients.Figure 8 shows the isentropic efficiency η of the S-CO2 centrifugal compressor at different flow coefficients.The normalized isentropic efficiency η/ηdesign, where ηdesign is the isentropic efficiency at the design point, exhibits a similar trend as the total pressure ratio when the flow coefficient ϕ decreases.In particular, at the operating point with the lowest flow coefficient ϕ, the isentropic efficiency of S-CO2 centrifugal compressor η reduces.This trend in Figure 8 suggests the flow loss mentioned above, which can worsen at lower flow coefficients and not only affect total pressure ratio, but also the isentropic efficiency η of the S-CO2 centrifugal compressor.

Flow Separation
As mentioned in section 3.1, with the decrease of the flow coefficient ϕ, a sort of flow distortion within the S-CO2 centrifugal compressor will expand and bring lower aerodynamic performance.The section describes a specific type of flow loss within the S-CO2 centrifugal compressor that results in lower aerodynamic performance.
With the aim of exploring the flow phenomenon inside the S-CO2 centrifugal compressor, Figure 9 compares the relative Mach number distributions at mid span cross section of the impeller domain at different flow coefficients.Relative Mach number, a dimensionless quantity defined as the ratio of the relative flow velocity to the local sound speed, provides the information of the fluid compressibility and the speed at which disturbances propagate through the fluid.Flow separation is indeed a phenomenon that occurs when the relative flow velocities near the surface of an object or within a flow field decrease to the point where they approach zero or become very low.This can lead to adverse effects such as boundary layer separation and the formation of vortex.As the separation occurs, the relative Mach number also tends to approach zero or become very low, indicating that the flow is slowing down significantly.In the vicinity of the critical point, where the compressibility of the fluid CO2 undergoes significant changes, utilizing the relative Mach number proves to be a more fitting approach for delineating the flow characteristics of the centrifugal compressor.
According to Figure 9, as the flow coefficient ϕ attains large values, a minor zone of the flow separation becomes evident on the blade pressure surface near the trailing edge (TE) within the S-CO2 centrifugal compressor.Such flow separation was also discovered in the study by Hacks et al. [15], and it was pointed out that this kind of separation contribute to flow loss inside the S-CO2 centrifugal compressors.At lower flow coefficients, this separation bubble will gradually expand and spread towards the TE of the blade.When the flow coefficient reaches its lowest point, this flow separation extends to almost the entire downstream channel of the impeller domain.Therefore, the conclusion can be drawn that a lower flow coefficient ϕ in the S-CO2 compressor results in a heightened occurrence of separation bubble on the blade pressure surface downstream the impeller domain, thereby causing increased flow loss.To sum up, the decline in performance at the lowest flow coefficient can be attributed to a kind of separation occurring on the blade pressure surface near the TE.

Unsteady Characteristics
Since the flow separation near the trailing edge of the blade of the S-CO2 centrifugal compressor is the most aggravated under the condition with the lowest flow coefficient ϕ, this condition is selected in this paper for unsteady CFD numerical calculation to obtain the unsteady characteristics of this type of flow separation.The following discussion is based on this unsteady CFD simulation.
As the flow separation occurs on the blade pressure surface near the TE, a numerical probe is placed near the impeller outlet to monitor the pressure during unsteady calculation.The fast Fourier transform (FFT) result of the unsteady pressure measured by this numerical probe, that is, the spectrum diagram of the pressure near the impeller outlet, is shown in Figure 10.Excepted the impeller passing frequency, a harmonic frequency of about 33 percent of the impeller passing frequency is also clearly observed in Figure 10.The pressure amplitude at this frequency is about half of the pressure amplitude at the impeller passing frequency.According to the information from Figure 10, it is obvious that the flow separation near the trailing edge of the blade can result in a harmonic to the pressure near the impeller outlet.That is to say, there is some form of unsteady evolution of this kind of flow separation, the frequency of this unsteady evolution corresponds to the harmonic frequency of about 33 percent of impeller passing frequency.In order to analyze the unsteady flow characteristics inside the S-CO2 centrifugal compressor more intuitively, the circumferential angle of the compressor is defined as shown in Figure 11. Figure 12 presents the unsteady evolution of the separation bubble on the blade pressure surface near the TE in a rotation period.The flow separation here is directly characterized in terms of relative flow velocity on the streamwise cross sections downstream the impeller domain, because the boundary conditions are set to a constant value in the unsteady numerical calculation, limiting the compressibility of the fluid S-CO2.
It is known from Figure 12 that the zone of this flow separation expands rapidly during time step of 0°to time step of 60°and then shrink gradually until it rotates to time step of 240°.From time step of 240°to time step of 300°, the flow separation zone almost stays steady.This may be because it is far away from the volute tongue, and the cross-sectional area of the volute is relatively uniform.When rotating from time step of 300°to time step of 360°, that is, time step of 0°in the next period, the flow separation region begins to expand again, but the expansion is relatively slow.It is noticeable that the time of the expansion of the flow separation region lasts from time step of 300°in the previous rotation period to time step of 60°in the next rotation period.The expansion time is about one third of the rotation cycle, consistent with the harmonic with a frequency of about 33 percent of the impeller passing frequency in Figure 10, confirming that the separation behavior brings a pressure fluctuation near the blade TE, resulting in the flow loss hindering the performance increase the S-CO2 centrifugal compressor.In general, severe separation bubble inside the fluid domain will lead to the emergence of vortices.In order to investigate the vortex distribution and its unsteady evolution caused by the separation bubble mentioned above, a dimensionless parameter named normalized helicity H is employed in this paper, defined as Equation (2).
Where   is the vorticity, W   is the relative flow velocity.The normalized helicity H presents the cosine of the angle between the vorticity and the relative flow velocity.If the normalized helicity H is equal to the positive unity, it means that a clockwise vortex appears in the fluid domain, while a counterclockwise vortex happens as this parameter reaches the negative unity.For the flow inside the centrifugal compressor, the vortex exhibits a clockwise rotation on the blade suction surface, whereas the vortex on the blade pressure surface rotates counterclockwise, a characteristic caused by the impeller's rotation itself.
The normalized helicity distribution on the downstream cross sections inside the impeller domain during one rotation period of the S-CO2 centrifugal compressor is presented in Figure 13.At the time steps of 0°, 60°and 300°, the abnormal clockwise vortex emerges on the blade pressure surface.The region occupied by the clockwise vortex is larger as the position moves closer to the blade TE.The intensity of the clockwise vortex is most pronounced at the time step of 60°and least prominent at the time step of 300°, indicating that the vortex enlarges from time step of 300°in the previous rotation period to time step of 60°in the next rotation period, corresponding to the time steps of deterioration of flow separation mentioned in above.The unsteady characteristic of the clockwise vortex near the blade pressure surface illustrates that it is caused by the separation bubble occurring on blade pressure surface.What's more, compared to the region of the flow separation mention above, the region of the clockwise vortex on the blade pressure surface is in proximity to the outlet of the impeller domain.It can be deduced that the pressure fluctuation of the impeller outlet is directly led by the vortex developed by the separation bubble on blade pressure surface.Therefore, under low flow coefficient conditions, the primary manifestation of flow loss within the S-CO2 centrifugal compressor is the presence of the abnormal clockwise vortex on the blade pressure surface downstream of the impeller domain.It can be found from Figure 13 that the vortex spreads obviously fast between time step of 0°and time step of 60°, that is, the first 1/6 rotation period.The detailed process of the deterioration of this vortex is illustrated in Figure 14.With the rapid expansion of the clockwise vortex region inside the impeller domain, it will shed off from the blade pressure surface to the middle of the impeller domain.The shedding vortex stays in the middle of the impeller domain for a period of time.Meanwhile, it diffuses to fill the entire passage as the time step is close to 60°.Throughout the time steps depicted in Figure 14, the clockwise vortex adjacent to the blade pressure surface not only evolves in the temporal dimension but also expands towards the downstream region of the impeller domain in the spatial dimension.The unsteady performance deterioration resulting from the evolution in these two dimensions on the downstream flow of the impeller domain reaches the maximum at the time step of 60°.

Conclusions
In this paper, the flow characteristics inside a S-CO2 centrifugal compressor deigned by Tsinghua University is studied by CFD numerical calculation.Steady CFD simulation is carried out to obtain the performance of the S-CO2 centrifugal compressor, while unsteady CFD simulation is performed to get the details of the flow separation at low flow coefficient condition.The unsteady characteristics of the flow separation is analyzed.From the study the following conclusions can be drawn: a) Firstly, when the flow coefficient decreases, there is a flow loss inside the S-CO2 centrifugal compressor hindering the improvement of its performance.The performance of the S-CO2 centrifugal compressor tends to reduce at the lowest flow coefficient conditions.b) Secondly, the separation bubble on the blade pressure surface near the downstream the impeller domain predominantly contributes to the decline in aerodynamic performance of the S-CO2 centrifugal compressor at lower flow coefficients.c) Thirdly, the separation phenomenon on the pressure surface near the blade TE has a strong unsteady effect.This unsteady effect can result in a large pressure fluctuation at the impeller outlet which frequency is about 33 percent of the impeller passing frequency.d) Lastly, during some time steps lasted for about one third of the rotation period of the S-CO2 centrifugal compressor, the flow separation near the pressure side of the blade deteriorates, causing the clockwise vortex generated from the pressure side.The clockwise vortex sheds off towards the middle zone of the impeller domain and diffuses towards the impeller outlet, occupying almost the entire downstream passage of the impeller domain at the time step of 60°, consist to the main form of the flow loss inside the S-CO2 centrifugal compressor.

Figure 2 .
Figure 2. Result of grid independence validation.

Figure 3 .
Figure 3. Sketch of the meshes of the S-CO2 centrifugal compressor.

Figure 4 .
Figure 4.The y+ distribution of the S-CO2 centrifugal compressor.

Figure 6 .
Figure 6.Comparison of performance curve by UDE experiment and CFD calculation.

Figure 7 .
Figure 7.The total pressure ratio at various flow coefficients.

Figure 8 .
Figure 8.The total pressure ratio at various flow coefficients.

Figure 9 .
Figure 9. Relative Mach number on 0.5 span cross section at various flow coefficients.

Figure 10 .
Figure 10.FFT result for unsteady pressure near the impeller outlet.

Figure 11 .
Figure 11.Sketch diagram of the rotation direction and circumferential angle.

Figure 12 .
Figure 12.Unsteady evolution of relative flow velocity on the streamwise cross sections near the blade TE during one rotation period.

Figure 13 .
Figure 13.Unsteady evolution of normalized helicity on the streamwise planes near the blade trailing edge during one rotation period.

Figure 14 .
Figure 14.Unsteady evolution of normalized helicity on the streamwise planes near the blade trailing edge during the first 1/6 rotation period.

Table 1 .
Main geometries and operating conditions of the S-CO2 centrifugal compressor.