Analysis of the Design of a Poppet Valve by Transitory Simulation

energies

Article Analysis of the Design of a Poppet Valve by Transitory Simulation

Ivan Gomez 1, Andrés Gonzalez-Mancera 1 , Brittany Newell 2 and Jose Garcia-Bravo 2,* 1 Mechanical Engineering Department, Universidad de Los Andes, Bogota 111711, Colombia; [email protected] (I.G.); [email protected] (A.G.-M.) 2 School of Engineering Technology, Purdue University, West Lafayette 47907, IN, USA; [email protected] * Correspondence: [email protected]; Tel.: +1-765-494-7312

Received: 10 February 2019; Accepted: 2 March 2019; Published: 7 March 2019

Abstract: This article contains the results and analysis of the dynamic behavior of a poppet valve through CFD simulation. A computational model based on the finite volume method was developed to characterize the flow at the interior of the valve while it is moving. The model was validated using published data from the valve manufacturer. This data was in accordance with the experimental model. The model was used to predict the behavior of the device as it is operated at high frequencies. Non-dimensional parameters for generalizing and analyzing the effects of the properties of the fluid were used. It was found that it is possible to enhance the dynamic behavior of the valve by altering the viscosity of the working fluid. Finally, using the generated model, the influence of the angle of the poppet was analyzed. It was found that angle has a minimal effect on pressure. However, flow forces increase as angle decreases. Therefore, reducing poppet angle is undesirable because it increases power requirements for valve actuation.

Keywords: poppet valve geometry; CFD valve simulation; valve optimization

1. Introduction An Electro hydraulic Servo Valve (EHSV) is a directional valve capable of accurately and proportionally metering flow based on a given input. These valves have been used extensively in aerospace applications since the 1950s particularly for flight control surfaces and other secondary functions. Other typical applications include power steering actuators and in electronically controlled industrial applications [1]. These valves are favored because they can be used to implement excellent feedback control systems while operating with relatively low power. In recent years model based diagnostics and prevention has been investigated to improve the reliability of Electro-hydraulic, Electro-mechanical and Electro-pneumatic systems [2–7]. In typical EHSV’s the radial clearance between the spool and the sleeve in the valves is in the order of 5 µm or 0.2 thousand of an inch [8]. This tight gap tends to break the fluid and form sticky films increasing viscous friction and eventually creating spool stiction. Finally, from a hydraulics perspective, systems operating with EHSV tend to be inherently energy inefficient because they restrict the flow passing from the supply port to the work port through metered orifices. This restriction creates a pressure drop that is converted into heat [9,10]. The heated fluid can vaporize at low pressure, increasing the possibility of cavitation in standard applications. Possible solutions to replace the traditional 4/3 directional controlled valve used in hydraulic systems to control the motion of an actuator include: (1) Displacement control actuation [11,12], (2) independent metering control [13,14], and (3) digital hydraulics [15–18]. Each of these solutions has its own strengths and weaknesses. While the first approach is great for controllability, stability, and efficiency, its implementation is costly because it requires a single pump for each actuator. Solution

Energies 2019, 12, 889; doi:10.3390/en12050889 www.mdpi.com/journal/energies Energies 2019, 12, 889 2 of 18

(2), improves energy efficiency significantly but still uses valves with spools that are also sensitive to contamination. Solution (3) uses poppet on-off valves (valves with only two positions), which are resilient against contamination, are relatively inexpensive and easy to operate. However, on-off poppet valves lack the speed of EHSV and the implementation of control strategies to achieve closed loop control of an actuator can be challenging. Solenoid operated poppet valves have been implemented in a great variety of applications. Thanks to their low cost and simple operation, these valves have replaced other technologies. For instance, the use of poppet style valves has been introduced in the automotive market replacing traditional cam-lever designs for controlling intake and exhaust in diesel engines [19]. Multiple solenoid poppet valves can be connected in parallel to construct a digital flow control unit (DFCU) to replicate the operation of an EHSV, with similar or better flow capacities and in theory better controllability, optimized energy consumption and significantly lower cost. With the purpose of replacing an EHSV for a group of DFCU’s, various researchers have focused on improving the resolution and controllability of an optimized number of valves [20]. One such control method is the implementation of Pulse Width Modulation (PWM) for controlling the flow using independent valves [21]. However, the typical high non-linearity of flow through an orifice in any valve still remains a great challenge. It has been proposed that a main issue of these non-linearities is their effect on response time of the valve being far greater than the control signal [22]. In other words, to improve the controllability issues due to non-linearities, it is necessary to increase the speed of the DFCU to obtain a comparable behavior to an EHSV. For poppet valves, the response time is always tied to the fluid flow forces, the slower the response time of the valve, the larger the flow forces acting on the poppet. One possible alternative to overcome these flow forces is to improve the electromagnetic properties of the solenoid operating the poppet valve. Some researchers have used solenoids with metal alloys like Al-Fe [19], Ti [20] or have resorted to the use of magneto-rheological fluids [23], with response times between 0.16 to 0.45 ms, 7–10 ms and 0.1–100 ms, respectively. Another parameter of interest for improving the response time of a DFCU is the geometry of the poppet and the seat. Studies have focused on decreasing the flow forces through empirical and computational analysis to evaluate the effect of angle on spool valves. Flow forces seen in these studies were reduced by approximately 36% [24]. Others have optimized rotary spool valves using genetic algorithms, where the energetic performance of the valve was improved by up to a 97% [25]. Computational Fluid Dynamics (CFD) is generally used for modeling and approximating the Navier-Stokes Equations to characterize flow through a valve. CFD analysis performed on poppet valves has been used to predict the behavior of these valves with relative errors below 8% [26,27]. Nevertheless, it is important to note that is has also been demonstrated that errors are related to many factors, such as geometry and that some configurations are more challenging for the mathematical models [28]. It is for this reason it is important to perform careful mesh sensitivity analysis to assess the performance of the model. The goal of this study was to characterize the behavior of a commercial poppet valve in transient state using CFD, and to focus on its response at high frequencies by analyzing the flow forces, geometric characteristics and pressure gradients. The model was validated using the manufacturer’s published experimental data and subsequently used to study the effect of geometric changes on the velocity of the poppet to make PWM control easier to implement on DFCU valves.

2. Problem Deﬁnition

2.1. Computational Domain The computational model was established starting with a commercial model of a solenoid controlled, 2 position and 3 port, poppet valve as seen in Figure1. A computational domain to characterize the flow through ports 1 and 2 was developed. Port 3 was not completely modeled because the disregarded portion doesn’t significantly affect the flow in the head area and therefore it is considered negligible. The cartridge valve was assumed to be inserted into a manifold where each port is connected to pipes assumed to be long enough to consider fully developed flow. The length of Energies 2019, 12, 889 3 of 18

Energies 2018, 11, x FOR PEER REVIEW 3 of 18 the pipes was estimated using turbulent ﬂow in pipe theory per Equation (1) and the properties of Energies 2018, 11, x FOR PEER REVIEW 3 of 18 95 flowﬂow at thethe normalnormal valvevalve operating operating conditions. conditions. It It was was estimated estimated that that the the pipe pipe length length should should not not be less be 96 lessthan than 0.13 0.13 m. m. 95 flow at the normal valve operating conditions. It was estimated that the pipe length should not be 1 𝟏 7𝟕 96 less than 0.13 m. 𝑳L𝒅d=𝟒.𝟒∙𝑫∙𝑹𝒆= 4.4·D·Re 6 , 𝟔, Re𝑹𝒆≤ 10 𝟏𝟎 (1)(1)

𝟏 𝟕 𝑳𝒅 =𝟒.𝟒∙𝑫∙𝑹𝒆 𝟔, 𝑹𝒆 𝟏𝟎 (1)

C.V. C.V.

Figure 1. Representation of the implemented valve: (a) ISO schematic representation; (b) picture Figureof the 1.valveRepresentation cartridge; (c of) detailed the implemented Section using valve: a CAD (a) ISO model. schematic representation; (b) picture of the Figure 1. Representation of the implemented valve: (a) ISO schematic representation; (b) picture valve cartridge; (c) detailed Section using a CAD model. 97 The ofgeometric the valve modelcartridge; of (thec) detailed poppet Section valve using was asimp CADlified model. by not considering elements like the 98 seals Theand geometricretaining modelrings from of the the poppet cartridge valve inserted was simpliﬁed in the manifold, by not considering as well as elements the fittings like theused seals to 97 The geometric model of the poppet valve was simplified by not considering elements like the 99 connectand retaining the tubbing rings fromto the the manifold cartridge ports. inserted These in elem the manifold,ents are assumed as well as to the have ﬁttings a negligible used to effect connect as 98 seals and retaining rings from the cartridge inserted in the manifold, as well as the fittings used to 100 wasthe tubbingnoted in to previous the manifold studies ports. [29, These30]. The elements geometry are of assumed the implemen to haveted a negligible domain is effect shown as was in Figure noted 99 connect the tubbing to the manifold ports. These elements are assumed to have a negligible effect as 101 2,in below. previous studies [29,30]. The geometry of the implemented domain is shown in Figure2, below. 100 was noted in previous studies [29,30]. The geometry of the implemented domain is shown in Figure 101 2, below.

102 Figure 2. Scaled down representation of the physical model domain at 45◦. 103102 Figure 2. Scaled down representation of the physical model domain at 45°.

° 104103 2.2. Governing EquationsFigure 2. Scaled down representation of the physical model domain at 45 .

105104 2.2.1.2.2. Conservation Governing Equations Equations 105 2.2.1. Conservation Equations

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2.2. Governing Equations

2.2.1. Conservation Equations The mass and moment conservation Equations for incompressible ﬂow excluding all external forces, including gravitational forces are given by the following Equations. It is assumed that changes in density of the ﬂuid are very small and therefore negligible in the analysis.

∇·U = 0 (2) and, δ ρ (U) + ∇·(U × U) = −∇p + ∇·τ (3) δt Assuming isentropic and Newtonian ﬂow, the stress tensor is deﬁned as: τ = µ ∇ × U + (∇ × U)T (4)

2.2.2. RANS Models The Reynolds-averaged Navier–Stokes (RANS) models are Equations of motion for fluid flow averaged over time and were used to model turbulence. The model assumes that an instantaneous quantity is decomposed into its time-averaged and fluctuating quantities.

U = U + Ue, p = p + pe, τ = τ + τe (5)

The RANS Equations for the conservation Equations become:

∇·U = 0 (6) δ ρ U + ∇· U × U = −∇p + ∇· τ − ρ Ue × Ue (7) δt where the term ρ Ue × Ue is known as the Reynolds stress. In this model the Boussinesq eddy viscosity assumption is used, and the Reynolds stress then becomes:

T 2 ρ Ue × Ue = µt ∇ × U + ∇ × U − ρIκ (8) 3 where µt is known as the turbulent viscosity or Eddy viscosity and κ is the average turbulent energy. The set of Equations (5)–(8) together with a turbulence model to solve for the turbulent viscosity are used to model the turbulent ﬂow and were solved using a commercial CFD solver (ANSYS®Academic Research Fluent, Release 17).

2.2.3. Wall Effect Functions Turbulent flows are significantly affected by the presence of walls where no slip occurs. This generates changes in the behavior of the turbulent flow and large velocity gradients, where generally speaking, linearized discretization methods do not appropriately match the real flow values. When using RANS models, particularly of the family of κ-ε models in this work, empirical functions known as wall effect functions may be implemented to account for the effects of this boundary layer. It is always necessary to develop a mesh with refined elements at and near the surface to satisfy boundary layer requirements. The level of refinement required depends upon both the turbulence model and the wall function being selected. Energies 2019, 12, 889 5 of 18

The standard wall effect function assumes a logarithmic proﬁle to model effects of turbulent ﬂow near a boundary. 1 1 U+ = lny+ + C = lnEy+ (9) k k where the terms k, E, C are empirical terms, U+ and y+ are non-dimensional terms for the velocity andEnergies the position2018, 11, x FOR of the PEER boundary REVIEW layer, respectively. 5 of 18

136 2.3.2.3. Numerical Numerical Methods Methods

137 2.3.1.2.3.1. CFD CFD Methods Methods 138 ForFor this this project project the the simulationssimulations werewere performedperformed usingusing ANSYSANSYS®®Academic Academic Research Research Fluent, Fluent, 139 ReleaseRelease 17. 17. InIn allall simulatedsimulated casescases thethe modelmodel waswas initializedinitialized usingusing potentialpotential flow,flow, the the SIMPLE SIMPLE 140 algorithmalgorithm was was used used for for the the solution solution of of the the conservation conservation Equations Equations and and second-order second-order discretization discretization 141 schemesschemes for for pressure, pressure, velocity velocity and and turbulent turbulent terms terms were were used. used. Finally, Finally, transient transient simulations simulations with with a a 142 temporallytemporally implicit implicit scheme scheme were were implemented implemented for for the the dynamic dynamic analysis. analysis. Results Results from from steady steady state state 143 simulationssimulations were were used used to initialize to initialize the model the andmo thedel “layering”and the meshing“layering” technique meshing was technique implemented was 144 forimplemented simulating thefor movementsimulating ofthe the movement valve poppet of the within valve thepoppet mesh. within the mesh. 145 TheThe constant constant pressurepressure outlet boundary conditions conditions were were implemented implemented at atthe the Section Section next next to port to 146 port3 and 3 andat the at exit the pipe. exit pipe.At the Atinlet the of inletthe pipe of the a fix pipeed velocity a fixed was velocity imposed was for imposed steady forstate steady simulations state 147 simulationsand a fixed and pressure a fixed condition pressure was condition imposed was for imposed the transient for the simulations transient to simulations ensure mathematical to ensure 148 mathematicalstability. An stability.interface An boundary interface boundarycondition conditionbetween betweenthe exit thepipe exit and pipe port and 2 port of the 2 of valve the valve was 149 wasimplemented, implemented, where where in case in case of ofmutual mutual contact contact one one side side would would act act as as an an internal internal face face and and the the other other as 150 asa aboundary boundary wall, wall, the the implemented implemented boundary boundary conditions conditions are are shown shown in in Figure Figure 3.3.

151 152 FigureFigure 3. 3.Diagram Diagram of of implemented implemented boundary boundary conditions. conditions.

153 2.3.2.2.3.2. Mesh Mesh Independence Independence 154 AllAll meshes meshes were were generated generated using using ANSYS ANSYS®®Academic Academic Research Research ICEM, ICEM, Release Release 17 17 software, software, the the 155 meshingmeshing method method used used is calledis called “Octree” “Octree” with with prismatic prisma elementstic elements generated generated for the for walls the walls at the at inlet the andinlet 156 outletand outlet pipes. pipes. The mesh The wasmesh reﬁned was refined at the valve,at the especiallyvalve, especially near the near poppet the poppet and seat and areas. seat Theareas. mesh The 157 wasmesh initially was initially generated generated assuming assuming ports 1 and ports 2 were 1 and closed, 2 were and closed, the “layering” and the tool“layering” for generation tool for 158 ofgeneration dynamic meshesof dynamic of ANSYS meshes®Academic of ANSYS® Research Acad Fluent,emic Research Release Fluent, 17 was usedRelease to connect17 was portsused 1to 159 andconnect 2 according ports 1 toand the 2 poppetaccording dimensions to the poppet as shown dime innsions Figure as4 .shown The independence in Figure 4. The of the independence mesh was 160 evaluatedof the mesh with was 5 progressively evaluated with reﬁned 5 progressively meshes applied refined over meshes the entire applied domain. over the entire domain.

161 162 Figure 4. Detail of elements generated using the “layering” tool in ANSYS.

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136 2.3. Numerical Methods

137 2.3.1. CFD Methods 138 For this project the simulations were performed using ANSYS® Academic Research Fluent, 139 Release 17. In all simulated cases the model was initialized using potential flow, the SIMPLE 140 algorithm was used for the solution of the conservation Equations and second-order discretization 141 schemes for pressure, velocity and turbulent terms were used. Finally, transient simulations with a 142 temporally implicit scheme were implemented for the dynamic analysis. Results from steady state 143 simulations were used to initialize the model and the “layering” meshing technique was 144 implemented for simulating the movement of the valve poppet within the mesh. 145 The constant pressure outlet boundary conditions were implemented at the Section next to port 146 3 and at the exit pipe. At the inlet of the pipe a fixed velocity was imposed for steady state simulations 147 and a fixed pressure condition was imposed for the transient simulations to ensure mathematical 148 stability. An interface boundary condition between the exit pipe and port 2 of the valve was 149 implemented, where in case of mutual contact one side would act as an internal face and the other as 150 a boundary wall, the implemented boundary conditions are shown in Figure 3.

151 152 Figure 3. Diagram of implemented boundary conditions.

153 2.3.2. Mesh Independence 154 All meshes were generated using ANSYS® Academic Research ICEM, Release 17 software, the 155 meshing method used is called “Octree” with prismatic elements generated for the walls at the inlet 156 and outlet pipes. The mesh was refined at the valve, especially near the poppet and seat areas. The 157 mesh was initially generated assuming ports 1 and 2 were closed, and the “layering” tool for 158 generation of dynamic meshes of ANSYS® Academic Research Fluent, Release 17 was used to 159 connectEnergies 2019 ports, 12, 8891 and 2 according to the poppet dimensions as shown in Figure 4. The independence6 of 18 160 of the mesh was evaluated with 5 progressively refined meshes applied over the entire domain.

161 162 Figure 4. DetailDetail of of elements elements generated usin usingg the “layering” tool in ANSYS.

Mesh reﬁnements were realized using approximately 20% increases in the number of elements between meshes. Simulations were evaluated at a velocity of 8.83 m/s, equivalent to approximately 4.44E-04 m3/s (7.04 gpm or 26.6 lpm), with ﬂuid properties as listed in Table1.

Table 1. Properties of the ﬂuid used in the simulation.

Material Temperature (◦C) Density (kg/m3) Dynamic Viscosity (kg·m/s) Oil AW 32 49 847.8 0.0195

Mesh independence was tested using two thousand iterations of the steady state simulations where the residuals were verified to be less than 10−5. For this mesh independence study standard κ-ε was used. The poppet has the standard commercial geometry and the state of the valve is fully open allowing flow from port 1 to port 2. The convergence parameter selected was a pressure differential measured at 5 diameters downstream and 10 diameters upstream with respect to the valve in accordance with ISO standard ISO 4411:2008 [31]. The pressure differentials obtained with the simulations were compared to the reported values from Sun hydraulics manufacturer’s data-sheet for valve model DWDA. All valves are tested at the manufacturer’s facility using petroleum-based hydraulic fluid with a viscosity of approximately 25 cSt. and a 19/17/14 fluid cleanliness level per ISO 4406 [32]. Table2 below presents a comparison between the published data and the corresponding simulated values.

Table 2. Summary of results for mesh convergence tests.

Elements Published Pressure (kPa) Simulated Pressure (kPa) Percent Error 3,970,308 540 603 11.70% 5,512,452 540 588 8.98% 8,239,820 540 576 6.72% 9,864,461 540 576 6.96% 12,308,409 540 574 6.45%

Results with errors less than 7% were obtained using the three ﬁner meshes, and based on these preliminary results, it was decided to use a mesh with approximately 9.8 million elements to ensure good resolution at a reasonable computational time in comparison to the mesh with 12.3 million of elements.

2.3.3. Turbulence Models Similar to the process followed to test the mesh convergence, an evaluation process for validating the turbulence models and wall functions was used. The evaluated turbulence models were the Energies 2019, 12, 889 7 of 18

Standard κ-ε, Realizable κ-ε, κ-ω SST and Spalart-Allmaras. Two wall functions were evaluated, mainly the Scalable Wall Functions (SWF) and Enhanced Wall Treatment (EWT) both available in the software. Various combinations of turbulence models and wall functions were tested for runs consisting of two thousand iterations of the steady state model and the pressure differentials obtained from the simulation were compared to that of the published results. Table3 below lists the results obtained for the selection of the most appropriate turbulence and wall function combinations. The results had a relative error of less than 13% for all turbulence models. The Standard κ-ε turbulence model with EWT was selected because it had a lower relative error.

Table 3. Summary of results for turbulence and wall function tests.

Turbulence Model Published Pressure (kPa) Simulated Pressure (kPa) Percent Error Standard κ-ε 540 577 6.96% Standard κ-ε SWF 540 531 1.50% Standard κ-ε EWT 540 543 0.61% Realizable κ-ε SWT 540 484 10.34% Realizable κ-ε EWT 540 550 1.94% κ-ω SST 540 607 12.6% Spalart-Allmaras 540 577 3.45%

Given the results presented in Table3 the combination for turbulence model and wall function of Standard κ-ε and EWT, respectively, is used in the rest of the work. For this ﬂow rate an error of less than 1% was obtained with respect to the published pressure drop. This combination of models is able to accurately model the boundary layer and the viscous dissipation therein.

3. Results and Discussion

3.1. Results Validation using Manufacturer’s Published Data Before proceeding to analyze the dynamic simulation results, the selected model was validated using the pressure differential at various flow rates. The simulations were developed in steady state with six thousand iterations obtaining residuals of magnitudes less than 10-6. These results are listed in Table4 below and plotted in Figure5. Notice that the slight difference between the pressure drop at the higher flow rate of Table4 and that obtained in the previous Section (Table3) is due to the results in this Section being obtained with six thousand iterations (two thousand in the previous Section). The simulated data demonstrates less than 5% error at flow rates of 2.24 × 10−4 m3/s (3.55 gpm or 13.4 lpm) and greater. The higher percentual error at lower flow rates is very likely due to the flow transitioning to laminar at this lower Reynolds numbers (approximately Re = 467 based on the diameter of the port for the lower flow rate) and the turbulence model struggling to accurately capture the physics of the system.

Table 4. Summary of pressure differential from static simulation at various ﬂow velocities.

Flow Rate (m3/s) Published Pressure (kPa) Simulated Pressure (kPa) Percent Error 4.44 × 10−4 540 559 3.59% 3.76 × 10−4 389 404 3.96% 3.01 × 10−4 252 262 4.08% 2.24 × 10−4 144 148 3.07% 1.47 × 10−4 64 67 5.42% 6.75 × 10−5 14 16 14.83% Energies 2019, 12, 889 8 of 18 Energies 2018, 11, x FOR PEER REVIEW 8 of 18

FigureFigure 5. 5.Comparison Comparison between between pressure pressure differential differential from from the the simulation simulation and an thed manufacturer’sthe manufacturer’s data. data. 3.2. Dynamic Analysis

3.2.1. Results at a Frequency of 25 Hz 214 3.2. Dynamic Analysis Starting from the good fitting of the simulation model and the manufacturer’s experimental data 215 sheet,3.2.1. Results the movement at a Frequency of the valve of 25 inHz. the dynamic analysis was modeled in three different ways. First, 216 for theStarting opening from process the good where fitting ports of 1the and simulation 2 are initially model disconnected, and the manufactur the modeler’s wasexperimental configured data so 217 thatsheet, the the displacement movement of of the the valve valve in was the representeddynamic anal byysis the was function modeled in Equation in three (10).different ways. First, 218 for the opening process where ports 1 and 2 are initially disconnected, the model was configured so P(t) = ±(0.5A − 0.5Atanh(K (t − D ))), K = 180, D = 0.2 (10) 219 that the displacement of the valve was representedh by thee functionh in Equatione 10. Equation (10) is a continuous approximation of the “head-viside” function, where A is the total 𝑃𝑡 =±0.5𝐴 −0.5𝐴𝑡𝑎𝑛ℎ𝐾𝑡−𝐷 , 𝐾 = 180, 𝐷 =0.2 (10) motion amplitude of the valve with a magnitude of 2.501 mm, Kh is a constant value affecting the time 220 requiredEquation for the (10) poppet is a continuous to traverse approximation from one position of the to “head-viside” the other, De isfunction, an induced where valve A is delay the total for 221 obtainingmotion amplitude steady state of the data valve before with the a poppet magnitude starts of moving. 2.501 mm, These Kh values is a constant were selected value affecting to match the 222 actualtime required valve time for the of 40 poppet ms required to traver tose move from the one poppet position from to the the other, fully closedDe is an to induced fully open valve position. delay 223 Thefor obtaining configuration steady for state the displacementdata before the of poppet the poppet starts for moving. opening These and closingvalues ofwere the selected valve assumed to match it 224 wasthe actual the same valve function time withof 40 anms opposite required sign. to move The control the poppet signal tofrom operate the fully the position closed to of thefully poppet open 225 wasposition. assumed The configuration to follow a sinusoidal for the displacement signal of the form:of the poppet for opening and closing of the valve 226 assumed it was the same function with an opposite sign. The control signal to operate the position of π 227 the poppet was assumed to follow a sinusoidalP(t) = Asin signal2π off + the form: (11) 2 𝜋 𝑃𝑡 = 𝐴𝑠𝑖𝑛 2𝜋𝑓 + (11) The implemented movements of the poppet are displayed2 in Figure6. 228 AllThe theimplemented simulations movements were initialized of the po withppet 6000 are iterationsdisplayed inin steadyFigure state6. at the initial position. Later the simulation was carried out in transient state with a simulation time of 0.08 ms and time steps of 0.001 ms and 30 iterations per time step. The simulations were evaluated using the optimal values obtained from the mesh independence study presented in Section 2.3.2., at approximately the same flow rate used in the study. The resulting inlet velocity was estimated to be 8.83 m/s and 666.4 kPa. The results of the dynamic analysis for opening and closing the valve are shown in Figure7. It was found that pressure drop during the single opening and closing procedure is similar to the result obtained with a fluctuating sinusoidal position command. Approximate values of 500 and 760 kPa were observed as maximum and minimum pressure drops during valve opening and closing

Energies 2018, 11, x FOR PEER REVIEW 9 of 18

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events. Small pressure instabilities during the opening and closing procedures were observed just after reaching the maximum position of the poppet while ports 1 and 2 were open. Changes to the time parameters De and Kh did not show a signiﬁcant effect on reducing or increasing the instabilities. Also, varying the amount of iterations per time step didn’t have much effect in reducing the instabilities. Therefore, these instabilities might be associated to the remeshing algorithm during the opening and 229 closing of the valve. Likewise, other authors have demonstrated that the force instability increases as 230 Energiesthe valve 2018, 11 opening, x FOR PEER is small REVIEWFigure which 6. might Transient suggest analysis that of the poppet instabilities displacement. are physical [33,34]. 9 of 18

231 All the simulations were initialized with 6000 iterations in steady state at the initial position. 232 Later the simulation was carried out in transient state with a simulation time of 0.08 ms and time 233 steps of 0.001 ms and 30 iterations per time step. The simulations were evaluated using the optimal 234 values obtained from the mesh independence study presented in Section 2.3.2., at approximately the 235 same flow rate used in the study. The resulting inlet velocity was estimated to be 8.83 m/s and 666.4 236 kPa. The results of the dynamic analysis for opening and closing the valve are shown in Figure 7. It 237 was found that pressure drop during the single opening and closing procedure is similar to the result 238 obtained with a fluctuating sinusoidal position command. Approximate values of 500 and 760 kPa 239 were observed as maximum and minimum pressure drops during valve opening and closing events. 240 Small pressure instabilities during the opening and closing procedures were observed just after 241 reaching the maximum position of the poppet while ports 1 and 2 were open. Changes to the time 242 parameters De and Kh did not show a significant effect on reducing or increasing the instabilities. Also, 243 varying the amount of iterations per time step didn’t have much effect in reducing the instabilities. 244 Therefore, these instabilities might be associated to the remeshing algorithm during the opening and 229 245 closing of the valve. Likewise, other authors have demonstrated that the force instability increases as 230246 the valve opening is smallFigure whichFigure 6. might 6.TransientTransient suggest analysis analysis that of of thepoppet poppet instabilities displacement. displacement. are physical [33,34].

231 All the simulations were initialized with 6000 iterations in steady state at the initial position. 232 Later the simulation was carried out in transient state with a simulation time of 0.08 ms and time 233 steps of 0.001 ms and 30 iterations per time step. The simulations were evaluated using the optimal 234 values obtained from the mesh independence study presented in Section 2.3.2., at approximately the 235 same flow rate used in the study. The resulting inlet velocity was estimated to be 8.83 m/s and 666.4 236 kPa. The results of the dynamic analysis for opening and closing the valve are shown in Figure 7. It 237 was found that pressure drop during the single opening and closing procedure is similar to the result 238 obtained with a fluctuating sinusoidal position command. Approximate values of 500 and 760 kPa 239 were observed as maximum and minimum pressure drops during valve opening and closing events. 240 Small pressure instabilities during the opening and closing procedures were observed just after 241 reaching the maximum position of the poppet while ports 1 and 2 were open. Changes to the time 242 parameters De and Kh did not show a significant effect on reducing or increasing the instabilities. Also, 243 varying the amount of iterations per time step didn’t have much effect in reducing the instabilities. 244 Therefore, these instabilities might be associated to the remeshing algorithm during the opening and 245 closing of the valve. Likewise, other authors have demonstrated that the force instability increases as 246 the valve opening is small which might suggest that the instabilities are physical [33,34].

FigureFigure 7. DynamicDynamic analysis analysis of pressure of pressure differential differential for opening, for closing opening, procedures closing and procedures fluctuating and ﬂuctuatingpositions. positions. 3.2.2. Parametric Analysis

The inlet pressure and input poppet frequency were changed to analyze the effect of these parameter on the behavior of the pressure drop, poppet force and ﬂow for two cycles of poppet motion using a sinusoidal input. These simulations were conducted in order to understand the effect on pressure differential at various poppet frequencies as well as different ﬂow inputs. The levels for each of the parameters used in the simulations are summarized in Table5. The time steps in the

Figure 7. Dynamic analysis of pressure differential for opening, closing procedures and fluctuating positions.

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247 3.2.2. Parametric Analysis 248 The inlet pressure and input poppet frequency were changed to analyze the effect of these 249 parameter on the behavior of the pressure drop, poppet force and flow for two cycles of poppet 250 motion using a sinusoidal input. These simulations were conducted in order to understand the effect 251 on pressure differential at various poppet frequencies as well as different flow inputs. The levels for 252 eachEnergies of2019 the, 12parameters, 889 used in the simulations are summarized in Table 5. The time steps in10 ofthe 18 253 simulations were adjusted so that at each step the poppet had achieved the same position irrespective 254 of the input frequency. The flow rates selected were based on simulations obtained during the steady simulations were adjusted so that at each step the poppet had achieved the same position irrespective 255 state validation of the mesh presented in Section 2. of the input frequency. The ﬂow rates selected were based on simulations obtained during the steady 256 state validation ofTable the 5. mesh Parameters presented and inlevels Section analyzed2. for the transient state simulations.

ParameterTable 5. Parameters1 and levels analyzed2 for the transient3 state simulations. 4 Frequency Parameter25 150 2 1003 4 200 (Hz) Inlet PressureFrequency (Hz) 25 50 100 200 Inlet Pressure (kPa)70.5 70.5271.7 271.7417.3 417.3 666.4666.4 (kPa)

257 Figure 88 displaysdisplays thethe pressurepressure gradientgradient forfor fourfour portionsportions ofof thethe cyclecycle whenwhen thethe poppetpoppet isis closingclosing 258 and opening using a sinusoidal input at 25 Hz. It was observed that during the closing process the 259 pressure difference between port 1 and the connectio connectionn to port 2 is greater than during the opening 260 process. The pressure drop, flow ﬂow forces on the poppet and flow ﬂow rates rates at at various various frequencies frequencies are are shown shown 261 inin Figure 99 forfor anan inletinlet pressurepressure ofof 666.4 666.4 kPa kPa and and several several poppet poppet frequencies. frequencies. TheThe pressurepressure dropdrop 262 increasedincreased for higherhigher cyclecycle frequencies,frequencies, as as did did the the forces forces on on the the poppet, poppet, especially especially during during the the closing closing of 263 ofthe the poppet. poppet. Peak Peak forces forces on on the the poppet poppet were were double double between between poppet poppet frequency frequency of 25of Hz25 Hz and and 200 200 Hz. 264 Hz.Flow Flow rates rates decrease decrease with higherwith higher poppet poppet frequencies freq anduencies with and respect with to respect the steady to statethe steady simulations, state 265 simulations,which was expected. which was expected.

Figure 8. Pressure gradients at various cycle timings for 25 Hz during closing and opening procedures. Figure 8. Pressure gradients at various cycle timings for 25 Hz during closing and opening Figureprocedures. 10 depicts the pressure gradient for frequencies between 25 Hz and 200 Hz at a timing of one a quarter cycle. It is worth noting that the pressure result at port 2 (outlet) seems to be unaffected by poppet frequency and increase pressure buildup is occurring mainly upstream.

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Figure 9. Simulated results at various frequencies for 666.4 kPa inlet pressure, (a) pressure differential, (b) net force on the poppet, (c) inlet flow, (d) outlet flow, (e) average inlet and outlet flows for second cycle. FigureFigure 9. Simulated9. Simulated results results at various at various frequencies frequencies for 666.4 for kPa666.4 inlet kPa pressure, inlet pressure, (a) pressure (a) differential,pressure 266 (Figureb)differential, net force10 depicts on(b) thenet the poppet,force pressure on the (c) inletpoppet,gradient flow, (c )for (inletd )freque outlet flow,ncies flow,(d) outlet between (e) averageflow, 25(e) inletHzaverage and and inlet200 outlet Hzand flowsat outlet a timing for of 267 one asecond quarterflows cycle.for cycle. second It cycle.is worth noting that the pressure result at port 2 (outlet) seems to be unaffected 268 by poppet frequency and increase pressure buildup is occurring mainly upstream. 266 Figure 10 depicts the pressure gradient for frequencies between 25 Hz and 200 Hz at a timing of 267 one a quarter cycle. It is worth noting that the pressure result at port 2 (outlet) seems to be unaffected 268 by poppet frequency and increase pressure buildup is occurring mainly upstream.

Figure 10. Pressure gradients during second cycle at 1.25 timing for various frequencies during the Figure 10. Pressure gradients during second cycle at 1.25 timing for various frequencies during the closing procedure. closing procedure. Figure 11 depicts the differential pressure and net force on the poppet and ﬂow rates at the inlet Figure 10. Pressure gradients during second cycle at 1.25 timing for various frequencies during the and outletclosing of portsprocedure. 1 and 2 when the valve is operated at 200 Hz and a range of inlet pressures. Similar

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269 Figure 11 depicts the differential pressure and net force on the poppet and flow rates at the inlet 270 and outlet of ports 1 and 2 when the valve is operated at 200 Hz and a range of inlet pressures. Similar 271 to the steady state behavior, it was observed that the pressure drop across the two ports increased 272 with increasing inlet pressure. This behavior is consistent for all studied frequencies. The results can 273 Energiesbe used2019 to, 12 estimate, 889 the required force necessary to close the poppet. It was observed that the12 force of 18 274 required to close the poppet increases sixfold between an inlet pressure of 70.5 kPa and 666.4 kPa. 275 The flow rate decreases as the frequency is increased. to the steady state behavior, it was observed that the pressure drop across the two ports increased 276 It can be observed in Figure 12 that pressure gradients decrease as inlet pressure decreases. The with increasing inlet pressure. This behavior is consistent for all studied frequencies. The results can 277 simulated pressure values for port 1 were between 70.5 kPa and 666.4 kPa. As inlet pressure increases be used to estimate the required force necessary to close the poppet. It was observed that the force 278 velocity also increases and the pressures in port 2 are maintained between 5 kPa and 20 kPa. At an 279 requiredinlet pressure to close of the666.4 poppet kPa, vortices increases were sixfold observed, between which an created inlet pressure low pressure of 70.5 points kPa andwith 666.4values kPa. of 280 The−5 ﬂowkPa to rate 15 decreaseskPa. as the frequency is increased.

FigureFigure 11. 11.Simulated Simulated results results at 200at 200 Hz Hz for fo varyingr varying inlet inlet pressures, pressures, (a) pressure(a) pressure differential, differential, (b) net(b) forcenet onforce the poppet, on the (cpoppet,) inlet flow, (c) inlet (d) outletflow, flow,(d) outlet (e) average flow, (e inlet) average and outlet inlet flowsand outlet for second flows cycle.for second cycle. It can be observed in Figure 12 that pressure gradients decrease as inlet pressure decreases. The simulated pressure values for port 1 were between 70.5 kPa and 666.4 kPa. As inlet pressure increases velocity also increases and the pressures in port 2 are maintained between 5 kPa and 20 kPa. At an inlet pressure of 666.4 kPa, vortices were observed, which created low pressure points with values of −5 kPa to 15 kPa.

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FigureFigure 12. 12.Pressure Pressure gradients gradients during during second second cycle cycle for thefor closingthe closing procedure procedure at varying at varying ﬂow ratesflow (ﬂowrates velocity). (flow velocity). 3.2.3. Non-dimensional Analysis

281 3.2.3.The Non-dimensional previous Section Analysis highlights the overall complexities involved in the interaction between all parameters involved in the valve dynamics at high frequencies. A non-dimensional analysis of the 282 The previous Section highlights the overall complexities involved in the interaction between all system behavior was performed for the purpose of obtaining a general representation of the results. 283 parameters involved in the valve dynamics at high frequencies. A non-dimensional analysis of the It is assumed that the pressure drop dp is a function of the valve geometry and the flow properties. 284 system behavior was performed for the purpose of obtaining a general representation of the results. Therefore the pressure drop can be expressed in terms of the relevant variables as: 285 It is assumed that the pressure drop dp is a function of the valve geometry and the flow properties. 286 Therefore the pressure drop can be expresseddp = g (inQ ,termsf , µ, ρ of, D the) relevant variables as: (12) 𝑑𝑝 =𝑔𝑄, 𝑓,𝜇,𝜌,𝐷 (12) where Q is the average flow rate, f is the poppet displacement frequency, µ is the viscosity of the fluid, μ 287 ρwhereis the densityQ is the of average the fluid, flow and rate,D is thef is diameterthe poppet at thedisplacement inlet for port frequency, 1. The non-dimensional is the viscosity analysis of the ρ 288 leadsfluid, to three is the dimensionless density of the numbers. fluid, and D is the diameter at the inlet for port 1. The non-dimensional 289 analysis leads to three dimensionless numbers. f ρD2 Qρ dpρD2 π = , π = , π = (13) f 𝜋 = , Q𝜋 = , 𝜋dp = 2 (13) µ µD µ 290 FigureFigure 13 13 shows shows the the results results of the of evaluation the evaluation of the non-dimensional of the non-dimensional numbers atnumbers the flow at conditions the flow 291 presentedconditions in thepresented previous in Section. the previous The curves Section. characterize The curves the behavior characterize of the non-dimensionalthe behavior of pressurethe non- 292 dropdimensionalπdp as a functionpressure ofdrop the non-dimensionalπdp as a function flowof the rate noπn-dimensionalQ for several frequencies flow rate relatedπQ for toseveral the 293 non-dimensionalfrequencies related number to theπ non-dimensionalf. number πf. Figure 13 shows that there is a direct relationship between πdp and πQ. Increasing the non-dimensional frequency number πf increases the slope in the πdp and πQ relationship. As the working frequency is increased the flow losses increase at a given flow rate. Numerically speaking, if the pressure drop is required to be kept constant at higher frequency of operation, it would be 7 necessary to decrease the flow rate, for instance for a constant πdp = 8 × 10 it would be required to decrease the flow rate by more than 50% to increase the frequency from 25 to 200 Hz. Using the non-dimensional numbers from Equation (13) and the plot in Figure 13, it could be inferred that implementing changes in some of the parameters of the valve or the operating conditions could lead to an improved valve dynamic behavior without incurring an increase in pressure loss. For example, decreasing the viscosity of the fluid is an effective way to be able to operate the valve at higher frequencies and flow rates due to the quadratic response between this constant and the πdp number. Increasing the diameter at port 1 would slightly decrease the flow and would increase moderately the pressure drop, at higher frequencies.

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294 295 Figure 13. Non-dimensionalized valve parameter relations.

296 Figure 13 shows that there is a direct relationship between πdp and πQ. Increasing the non- 297 dimensional frequency number πf increases the slope in the πdp and πQ relationship. As the working 298 frequency is increased the flow losses increase at a given flow rate. Numerically speaking, if the 299 pressure drop is required to be kept constant at higher frequency of operation, it would be necessary 300 to decrease the flow rate, for instance for a constant πdp = 8 × 107 it would be required to decrease the 301 flow rate by more than 50% to increase the frequency from 25 to 200 Hz. Using the non-dimensional 302 numbers from Equation (13) and the plot in Figure 13, it could be inferred that implementing changes 303 in some of the parameters of the valve or the operating conditions could lead to an improved valve 304 dynamic behavior without incurring an increase in pressure loss. For example, decreasing the 305 viscosity of the fluid is an effective way to be able to operate the valve at higher frequencies and flow 306 rates due to the quadratic response between this constant and the πdp number. Increasing the diameter 294307 at port 1 would slightly decrease the flow and would increase moderately the pressure drop, at higher 308 frequencies. 295 Figure 13. Non-dimensionalized valvevalve parameter relations.

309 3.2.4.3.2.4. EvaluationEvaluation ofof VariationVariation ofof thethe PoppetPoppet AngleAngle 296 Figure 13 shows that there is a direct relationship between πdp and πQ. Increasing the non- 297310 dimensionalTheThe seatseat frequency and poppet number angle angle wereπ weref increases another another the design design slope para in parameter themeter πdp usedand used π toQ torelationship.evaluate evaluate the the sensitivityAs sensitivity the working of the of 298311 thefrequencymodel. model. Angles is Angles increased lower lower than the than theflow commercial the losses commercial increase poppet poppetat werea given were tested flow tested to rate. study to Numerically study the effect the effect on speaking, pressure on pressure if drop the 299312 droppressureand flow and drop ﬂowforces is forces duringrequired during closing to be closing keptand openingconstant and opening events.at higher events. Figure frequency Figure 14 depicts of14 operatiodepicts various variousn, itrepresentations would representations be necessary of the 300313 oftoredesigned thedecrease redesigned thepoppet flow poppet rate,geometries. for geometries. instance The for Thechange a constant change in inangleπdp angle = 8 was× 10 was 7 itimplemented implementedwould be required insuring insuring to decreasethe poppetpoppet the 301314 displacementflowdisplacement rate by more remainedremained than 50% unchangedunchanged to increase andand the guaranteeingguaranteeing frequency fromthatthat the the25 clearancetoclearance 200 Hz. between betweenUsing the thethe non-dimensional poppetpoppet andand thethe 302315 seatnumbersseat waswas maintainedmaintainedfrom Equation toto minimize minimize(13) and the pressurepressure plot in losses lossesFigure at at 13, valve valve it could closing. closing. be inferred that implementing changes 303 in some of the parameters of the valve or the operating conditions could lead to an improved valve 304 dynamic behavior without incurring an increase in pressure loss. For example, decreasing the 305 viscosity of the fluid is an effective way to be able to operate the valve at higher frequencies and flow 306 rates due to the quadratic response between this constant and the πdp number. Increasing the diameter 307 at port 1 would slightly decrease the flow and would increase moderately the pressure drop, at higher 308 frequencies.

309 3.2.4. Evaluation of Variation of the Poppet Angle

310 The seat and poppetFigure angle 14. werePoppet another valve anglesdesign evaluated parameter in theused simulation. to evaluate the sensitivity of the 311 model. Angles lower thanFigure the 14. commercial Poppet valve poppet angles were evaluated tested in theto study simulation. the effect on pressure drop 312 and flowThe resultsforces during for the variationclosing and of theopening poppet events. angle Figure are depicted 14 depicts in Figure various 15 .representations It was found that of the angle has minimal effect on pressure differential, the maximum values obtained were between 107 kPa 313 redesigned poppet geometries. The change in angle was implemented insuring the poppet 314 todisplacement 110 kPa, which remained are less unchanged than a 1% and change guaranteeing with respect that to the the clearance original poppet between angle. the poppet and the 315 seat wasSimilarly, maintained changes to inminimize the flow pressure rate at the losses inlet at and valve outlet closing. where not significant. However, the flow forces during the entire motion of the poppet were seen to increase as the angles were decreased, which from an energetic perspective allows us to conclude that reducing the poppet angle is undesirable because it would lead to larger power requirements for actuation of the valve. Figure 16 shows the pressure gradients for the closing process, it can be seen from the Figure that as the angle is reduced, negative pressure zones were reduced, which is associated with larger changes in the flow direction at port 2, leading to accelerated flows during the closing of port 1. This is assumed to be the principal cause for the increase in flow forces for lower angles. This is considered to be a local phenomenon, meaning it only affects the velocity, pressure and force around the poppet, but does not affect global variables such as inlet and outlet flow and pressure differential.

Figure 14. Poppet valve angles evaluated in the simulation.

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316 The results for the variation of the poppet angle are depicted in Figure 15. It was found that the 317 angle has minimal effect on pressure differential, the maximum values obtained were between 107 318 kPa to 110 kPa, which are less than a 1% change with respect to the original poppet angle.

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316 The results for the variation of the poppet angle are depicted in Figure 15. It was found that the Energies 2019, 12, 889 15 of 18 317 angle has minimal effect on pressure differential, the maximum values obtained were between 107 318 kPa to 110 kPa, which are less than a 1% change with respect to the original poppet angle.

Figure 15. Simulated results for a 25 Hz operating frequency and a 666.4 kPa inlet pressure. (a) Pressure differential, (b) net force on the poppet, (c) inlet flow, (d) outlet flow.

319 Similarly, changes in the flow rate at the inlet and outlet where not significant. However, the 320 flow forces during the entire motion of the poppet were seen to increase as the angles were decreased, 321 which from an energetic perspective allows us to conclude that reducing the poppet angle is 322 undesirable because it would lead to larger power requirements for actuation of the valve. 323 Figure 16 shows the pressure gradients for the closing process, it can be seen from the Figure 324 that as the angle is reduced, negative pressure zones were reduced, which is associated with larger 325 changes in the flow direction at port 2, leading to accelerated flows during the closing of port 1. This 326 is assumed to be the principal cause for the increase in flow forces for lower angles. This is considered 327 to be a local phenomenon, meaning it only affects the velocity, pressure and force around the poppet, FigureFigure 15.15.Simulated Simulated results results for for a 25a 25 Hz Hz operating operating frequency frequency and aand 666.4 a 666.4 kPa inlet kPapressure. inlet pressure. (a) Pressure (a) 328 but doesdifferential,Pressure not affect differential, (b) global net force ( bvariables) onnet the force poppet, such on the as (c poppet,) inlet inlet ﬂow,and (c) outletinlet (d) outlet flow, flow ﬂow.(d and) outlet pressure flow. differential.

319 Similarly, changes in the flow rate at the inlet and outlet where not significant. However, the 320 flow forces during the entire motion of the poppet were seen to increase as the angles were decreased, 321 which from an energetic perspective allows us to conclude that reducing the poppet angle is 322 undesirable because it would lead to larger power requirements for actuation of the valve. 323 Figure 16 shows the pressure gradients for the closing process, it can be seen from the Figure 324 that as the angle is reduced, negative pressure zones were reduced, which is associated with larger 325 changes in the flow direction at port 2, leading to accelerated flows during the closing of port 1. This 326 is assumed to be the principal cause for the increase in flow forces for lower angles. This is considered 327 to be a local phenomenon, meaning it only affects the velocity, pressure and force around the poppet, 328 but does not affect global variables such as inlet and outlet flow and pressure differential.

FigureFigure 16. Simulated 16. Simulated pressure pressure gradient gradient for closing for closing event using event various using various poppet poppet angles. angles. 4. Conclusions

A model able to simulate the flow through a poppet valve was developed. The model was validated by comparing the pressure drop at different flow rates and comparing with data provided by the manufacturer of the valve. Values within 5% error were obtained for most flow rates. The model is also able to model the dynamic behavior of the valve as the poppet opens and closes cyclically at different frequencies. The model was used to characterize the behavior of the flow through the valve as frequencies increased from 25 Hz to 200 Hz. It was established that as frequency of operation of the valve increased flow rate decreased. Also, forces on the poppet, related to the force needed to actuate the valve, increased with increasing frequencies. For example, at the higher inlet pressure studied, force Figure 16. Simulated pressure gradient for closing event using various poppet angles. increased twofold as frequency went from 25 Hz to 200 Hz.

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In order to obtain more general results and to gain better insight on the process, a non-dimensionalized parametric study revealed that changes in the fluid’s viscosity or the diameter of the ports could effectively improve the performance of the valve. The analysis showed that as frequency increased, the slope of the relation between pressure drop and flow rate became steeper. This means that increasingly higher pressure drops occurred for a given flow rate as frequency increased. It was shown that pressure drop under particular conditions were proportional to the square of the viscosity and inversely proportional to the square of the diameter of the ports. Taking these parameters into consideration may allow to design poppet valves systems that are able to operate at higher frequencies. Finally, it was shown that poppet angle had little effect on the overall performance of the system. In order to continue improving the understanding of the overall behavior of poppet valves at high frequencies and improve the design of such valves to make them capable of operating at high frequencies, it would be interesting to couple the current model with that of the solenoid responsible for operating the valve. This will allow to incorporate the response of that system into the overall time response of the valve. The current model can also aid in analyzing novel designs that are able of reducing pressure drop and force on the poppet at high frequencies.

Author Contributions: Conceptualization, A.G.-M. and J.G.-B.; methodology, I.G., J.G.-B. and A.G.-M.; software, A.G.-M. and I.G.; validation, I.G., J.G.-B. and A.G.-M.; formal analysis, I.G.; investigation, I.G.; resources, I.G.; data curation, I.G.; writing—original draft preparation, I.G. and J.G.-B.; writing—review and editing, I.G., J.G.-B., B.N. and A.G.-M.; visualization, I.G.; supervision, A.G.-M. and J.G.-B. Funding: This research received no external funding. Conﬂicts of Interest: The authors declare no conﬂict of interest.

Nomenclature

A Total movement amplitude of the poppet C Empirical constant for the wall boundary De Valve delay dp Pressure drop across de valve E Empirical constant for wall boundary f Valve oscillation frequency I Turbulence intensity K von Karman constant Kh Velocity factor for the change of head-viside function P(t) Valve position function p, p, pe Total pressure, average and fluctuating Q Port 1 flow rate t Time U, U, Ue Total velocity, average and fluctuating U+ Non-dimensional velocity Y+ Non-dimensional distance to the wall κ Turbulence energy µ Dynamic viscosity µt Turbulence viscosity πdp Loss number πf Frequency number πQ Flow number ρ Fluid’s density τ, τ, τe Total Stress tensor, average, fluctuating Energies 2019, 12, 889 17 of 18

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