CFD AS A DESIGN TOOL FOR FLUID POWER COMPONENTS

Transcription

1 CFD AS A DESIGN TOOL FOR FLUID POWER COMPONENTS M. BORGHI - M. MILANI Diartimento di Scienze dell Ingegneria Università degli Studi di Modena Via Cami, 213/b Modena borghi@omero.dsi.unimo.it - milani@omero.dsi.unimo.it R. PAOLUZZI CEMOTER-CNR Via Canal Bianco, Cassana, Ferrara - Italy aoluzzi@cemoter.bo.cnr.it ABSTRACT The aer collects some examles of CFD alication to hydraulic comonents design, aimed at showing how a comlex tool can be used both for the internal flow characterisation and the macroscoic analysis of comonent behaviour. In its first art, the aer deals with the steady-state characterisation of a commercial 4/3 on/off distributor, and focuses the attention on CFD strategies used in order to obtain an accurate descrition of steady-state axial flow-forces, acting on the sool of the comonent. The solution of simlified models (axis-symmetric) and the develoment of an extended and novel non-dimensionalization technique are the key to the use of a limited number of distributor configurations and a minimum number of oerational conditions to obtain a full characterisation of the comonent. The second art of the aer is devoted to hydraulic locking analysis, where CFD is alied to taered 3D geometry in order to validate the results of secific design assumtions and to verify the results obtained by a simlified, though two-dimensional, aroach to radial forces estimation in sool valves. As a final examle, the last art of the aer is dedicated to article tracking analysis and collision/erosion rediction, using a simlified aroach based on the hyothesis of uncouled motion of fluid and articles. In this last case, the CFD comuted flow field, is used in order to allow the solution of the lagrangian equations of motion for solid articles dragged by the fluid according to a given law. The aim of the examles collected is not the solution of the roblems dealt with (to this urose the reader is referred to the bibliograhy), but rather the identification of areas and methods where the alication of a comlex and exensive design tool like CFD analysis can reach the breakeven oint of a convenient alication in the design hase of a hydraulic comonent. 1

2 HYDRAULIC DISTRIBUTOR Axial flow forces in sool tye valves are one of the most imortant, and less known, factors affecting their actual dynamic behaviour. The exerimental measurement is always difficult and must be erformed indirectly, to this urose an extensive exerimental characterisation (whose details are reorted in [2]) of the axial flow force has been erformed ([1], [2]),using the rototye of a 4/3 on/off distributor shown in Figure 1. The flow-rate across the distributor was rogressively increased in slow stes in order to collect values reresentative of a steady state oeration of the valve. The quantities measured included ressure at all ressurized orts, sool travel, overall axial force and flow rate. Figure 1 Prototye used for exerimental characterisation of a 4/3 on-off distributor. At the same time, a wide CFD exerimentation was lanned, considering four different sool ositions for an uncomensated ort connection. The values of the metering area oenings determined by the sool osition were evenly saced in the range of exerimental data collected. The oenings investigated where at 0.15, 0.25, 0.35, 0.5 and 0.6 mm of sool travel. Model simlification and non-dimensionalization Following the aroach detailed in [3] and given in Aendix, numerical models were dramatically simlified, describing the geometry as axis-symmetric and the fluid dynamic roblem as turbulent, isothermal, non-dimensional and involving a Newtonian and incomressible fluid. A further and much more imortant ste in the direction of roblem simlification is given by the non dimensionalization rocess introduced, aimed at extending the field of alication of a single analysis not only to the class of roblems considered similar (i.e. having the same Reynolds number) but also to the entire range of turbulent Reynolds number for roblems having geometric similitude only. The entire mathematical rocess is 2

3 fully described and alied in [3] and, to the urose of this examle, it is only worth recalling that: 1. The geometry must be made non dimensional with resect to the metering section oening; 2. The velocity scale of the roblem is the average velocity across the metering section, comuted for a reference flow rate used for Reynolds number definition; 3. The boundary conditions alied shall be described in terms of inlet velocity rofile; 4. The solutions obtained for a single (fixed Reynolds number) case can be extended to the full range of exected flow rate, as long as the conditions for a fully develoed turbulent efflux in the metering section are satisfied. This rocedure, though more demanding in the roblem set-u hase and less immediate in the ost-rocessing hase, dramatically imroves the amount of information which can be extracted by the designer from a single numerical run. CFD APPLICATION Figure 2 collects some examles of the internal flow field in the uncomensated connection. The figures shown are referred to velocity vector magnitude (10 contours, with non dimensional values between 0 and 1), resectively in the smallest and the largest among the oenings analyzed. These information, combined with ressure distributions inside the fluid domain, allow a thorough analysis of internal flow henomena (Coanda effects, streamlines, fluid-structure interaction, ) and the comutation of characteristic coefficients related to the steady state behavior of the distributor (discharge coefficient, hydraulic resistance, jet angle, ). a) 0.15 mm oening b) 0.6 mm oening Figure 2 CFD flow field Seed distributions inside uncomensated connections. 3

4 Secific attention was aid to the comutation of the axial flow force acting on the sool, aiming at a numerical-exerimental comarison in order to validate the model and to assess the degrees of aroximation imlied by the assumtions. The results of the study show how the axial flow force rediction made ossible by the use of CFD tools, reroduces with sufficient accuracy the exerimental evidence, and Figure 3 shows the numerical vs. exerimental data comarisons as obtained for the smallest and the largest oenings considered F (N) Exerimental Data (with regression curve) Numerical Results Re = bar a) 0.15 mm oening F (N) 150 Exerimental Data (with regression curve) Numerical Results 100 Re = bar b) 0.6 mm oening 4

5 Figure 3 Numerical vs. exerimental comarison of steady state axial flow force. For small oenings, where the flow can be considered fully turbulent also at low values of the flow-rate, and the ressure has, with a good degree of aroximation, an axis-symmetric distribution, the numerical rediction fits almost erfectly the exerimental data. Conversely, for oenings close to the sool travel end-sto, where the axis-symmetric aroximation of the geometric domain is less accurate, the CFD rediction of axial flowforce aears underestimated, with an error of the 10-15% with resect to exerimental evidence. The reasons of this underestimation are still to be fully exlained, esecially when the effect is considered together with the almost erfect reresentation of the ressure dro across the metering edge. Provided that some flow conditions are satisfied (namely a low Mach number), the data obtained on steady-state flow forces can be used to estimate the transient to stationary flow force ratio. For instance, in [4] it is shown how to estimate the transient flow force due to a slow variation in the valve flow rate at various oenings. It was found that, even in the case of quasi static flow, the dynamic flow force can count for several ercent of the total flow force comonent. According to the aroach reorted in [4], the ratio of dynamic to static force can be exressed as F F d 1 s L dq = v Q dt and the figure below gives an examle of the results that can be obtained by the extraolation of the stationary CFD results to the slow transient alication, where the ratio between the two flow force comonents is given as a function of the rate of change in flow (R, in l/min/s) and of the non dimensional ratio between daming length (L) and sool travel. From the observation of the surface it can be noticed that the transient effect becomes raidly more relevant than the stationary one. This is imortant esecially because they act in oosite directions. The knowledge of the ratio of transient to stationary flow force value allows, at least for relatively slow transient, a rough estimate of the final actual effect of the correction needed to correlate CFD stationary results to slow dynamic flow conditions as, for instance, caused by cyclic flow variation in circuits. 5

6 RADIAL FORCES ANALYSIS The axial forces are not the only one affecting the sool equilibrium. In many cases, radial forces generated by the sool dislacement and/or by its taering, may generate radial forces usually referred to as locking forces. These forces have been thoroughly investigated in the ast, and many aroaches exist to their comutation, but all of them more ore less share the common assumtion of a negligible circumferential flow rate, and therefore rely on a mono-dimensional analysis. The investigation of the flow field inside the clearance is erformed for different eccentric ositions of the sool with resect to the sleeve axis of revolution, and used to redict the total side thrust acting on the sool [5],. As in the revious one, also in this case the fluid dynamic roblem is stated as nondimensional and referred to a Newtonian and incomressible fluid, but the solution is erformed with reference to a laminar flow induced into the clearance by a given ressure difference alied to the boundaries. Figure 4 shows ressure distributions in a taered clearance when the sool is dislaced at a mid eccentric osition (a) and at an eccentricity close to the maximum (b). 6

7 Both distributions indicate clearly that no radial ressure gradient exists inside the clearance, but evidence how the ressure field is characterized by both an axial and an angular comonent. a) mid eccentricity b) maximum eccentricity Figure 4 CFD flow fields Pressure distributions locking the sool against the sleeve. These solutions directly give the total side thrust acting on the sool, and can be erformed in order to analyze the influence of geometric characteristics of the clearance (as the taer, the axial length and the radial height can be) [5], on locking force. In the aer, the CFD based redictions of locking force is comared with values obtained by design rocedures usually found in literature [7], [8], and develoed according to a simlified aroach. Figure 4-bis Percentage error in the comutation of locking force between simlified aroaches and two dimensional flow field solution (R indicates taer to clearance ratio) 7

8 The results shown in [5], and recalled in Figure 4-bis, indicate a significant error in locking force rediction associated to simlified aroach. These errors range from 30% u to 70% and deend on taered clearance descritive arameters. The CFD analysis, roerly combined with the continuity equation alied to the clearance volume, can also be used in the analysis of taered clearances with balancing grooves, in order to calculate the ressure value inside the grooves themselves and to investigate the influence of their geometry and osition in reducing the total side thrust. In [5], the first art of this analysis, related to rectangular grooves with shar edge, is resented, together with the comarison of groove ressure numerically redicted or exerimentally determined. To this urose, figure 5 shows the comarison made for single grooved istons, varying the groove length, its axial osition, and imosing different ressure boundary condition to the clearance; a sketch of the geometry considered with a list of symbols is also shown. It is worth noting that the exerimental-numerical comarison is almost erfect when the groove is located closer to the low ressure end of the sool, and the differences increase with the absolute ressure values. Figure 5 Influence of length and osition on ressure level inside a single balancing groove Gr 3 oov e Pre 2.5 ssu re - 2 MP a 1.5 L G = 2 mm LG = 4 mm L G = 4 mm L G = 4 mm Taered Piston with a Single Groove Different Groove Lenghts, L G Exerimental - 4 MPa Calculated - 4 MPa Calculated - 2 MPa Exerimental - 2 MPa 1 LG = 4 mm LG = 2 mm 0.5 LG = 4 mm LG = 2 mm Groove Centre Distance from High Pressure Boundary / Total Axial Length The following observations were made ossible by the investigation: it was confirmed that the groove reduces the entity of locking force; for the groove having length ten times the clearance this reduction seemed to be slightly deendent on the groove axial osition; 8

9 no more advantage seemed to be obtainable ositioning this groove at a distance from the high ressure side lower than L/4. In Table 1 the total locking force ercentage reductions are reorted, for different values of ercentage eccentricity of the sool axis. As suggested in [11], for both the grooves considered, the ercentage of reduction is calculated in resect of locking force acting on a simly taered sool. The trend suggested in [11] aears confirmed, even if the effect of a single groove on locking force reduction does not reach the values reorted therein. F % L G /C=10 Groove L G /C=30 Groove e % at L/2 at L/4 at L/2 at L/ Table 1: Total Locking Force Percentage Reduction for Different Groove Lengths and Positions. PARTICLE TRACKING The investigation of the motion of contaminant articles inside the fluid is the first ste to be erformed in order to tackle the comlex roblem of erosion effect on the service life of comonents. To this urose, the analysis of the two-hase flow interesting each hydraulic comonent or machine (fluid lus contaminant articles) is simlified into the uncouled solution of the Navier-Stokes Equation (for the fluid flow) and the LaGrange Equation of article motion, once solid articles are inserted into the fluid [9]. a) Eroded conical sool b) Numerical Prediction 9

10 Figure 6 Exerimental and numerical detection of growing contamination concentration and fatigue tests influence on a conical sool. The study was erformed for metallic articles having different average dimensions and different concentrations. A qualitative reresentation of actual fluid-structure interaction is ossible, and gives useful information for hydraulic comonent internal geometry design. The comutation of the trajectory of a article imlies the knowledge of its kinetic energy and imact angle on the surface. This last imact can be considered [10] either as elastic (analysed using the theoretical formulation coming from Hertz theory of contact), or inelastic. It is imortant to remind that imact characteristics of two bodies (imacted surface and solid article) coming in contact are functions of: article mass; article average dimensions; relative velocity of two bodies; Young and Poisson constants of surface and article materials; Yield stress of the imacted material. If the inelastic imact is considered, an elastic-lastic material reach the limit of elastic behaviour at a oint beneath the surface when the maximum contact ressure at the instant of maximum comression reaches the value 1.6Y, being Y the yield stress of the softer body. In case of a shere striking a lane surface of a large body in the normal direction, the minimum article velocity necessary to cause yielding can be exressed as [10]: V yielding Y 5 E 2 Y ρ (6) where ρ is the density of the imacting article and E is the equivalent Young modulus of the two bodies (article and surface) defined by: 1 E = 2 ( 1 ν ) ( 1 ) 2 ν E + E (7) The imact velocity causing yield in metal surfaces is very small: for examle, for an hard steel shere striking a medium hard steel ( Y 1000MPa ), it assumes the value of about 0.15 m/s. This estimation can not neglect, obviously, the influence of surface treatments, the actual geometry of solid contaminants, the imact direction and the actual elastic characteristics of imacting bodies, that lead to a higher value of the yielding imact velocity. On the other side, it is necessary to oint out that discharge henomena interesting hydraulic comonents lead to average velocities of the fluid at the metering edge in the range of 80/120 m/s (deending on metering oening and incoming flow rate), giving to the jet the necessary imulse to cause erosion. 10

11 Figure 6, for examle, comares the asect of a conical sool at the end of a fatigue test cycle, erformed for a growing concentration of contamination, with the article tracks obtainable using CFD. It is worth noting how, within the limits and limitations of the numerical aroach to the roblem, the CFD reresentation of articles motion seems to justify the location of zones interested by a high robability of article imact, and to redict the location and effect of erosion on the sool, at least qualitatively [9]. Comutational Fluid Dynamics has an aarently tremendous otential in this resect. The reliminary results indicate that a forecast of comonent service life is a feasible task, rovided that adequate consideration is also given to the erosion effect of other henomena, like for instance cavitation. A considerable amount of work is however still to be done in order to assess definitely the ractical usefulness of result obtained with the roosed aroach. CONCLUSIONS Comutational fluid dynamics is not to be considered a new alication in the field of fluid ower, but the general feeling about it it that it is a too comlex tool to be alied to low-cost comonents, where the geometrical shae introduces additional difficulties in a roblem whose mathematical formulation is undoubtedly comlex. Comuter technology advances and user friendly software have a tremendous otential in boosting the ower of this tool, but its cost is still considered too high for a generalized use at the design stage of hydraulic comonents. One ossible workaround to the roblem is the maximization of the amount of information extracted from every single run of the numerical model. This aroach has been shown through four different examles which demonstrate that: 1. A comrehensive aroach to roblem non-dimensionalization enables the analyst to use the result of a single numerical run to derive a full characteristic curve for a given valve geometric osition (oening); 2. Stationary analysis can be extraolated to slow transients in order to estimate the dynamic to stationary flow force ratio; 3. Radial forces effect (locking) can be investigated with a greater detail, and CFD can drive the set u of more reliable redictive formulas to be used in valve design; 4. Particle tracking techniques enable the use of CFD to the urose of erosion estimation, oening the gate for interesting develoments in the field of service life assessment and roactive maintenance of comonents and systems. The increased benefit coming from the concurrent use of all of the ossibilities shown, otentially change considerably the breakeven oint of convenience in the use of CFD as a tool integrated in the design rocess of a manufacturer, heling in the broadening of the fruition base of existing commercial rograms. ERENCES 1. Palumbo A., Paoluzzi R., Borghi M., Milani M. - Forces on a Hydraulic Valve Sool - Proceedings of the Third JHPS International Symosium on Fluid Power Pagg Yokohama, 4-6 November,

12 2. Borghi M., Cantore G., Milani M., Paoluzzi R. - Exerimental and Numerical Analysis of Forces on a Hydraulic Distributor - Proceedings of The Fifth Scandinavian International Conference on Fluid Power, SICFP 97 Vol. I Pagg Lingkoeing, Sweden, May, Borghi M., Cantore G., Milani M., Paoluzzi R. - Analysis of Hydraulic Comonents Using Comutational Fluid Dynamics Models - Paer Proceedings of the Institution of Mechanical Engineering Vol 212 Part C - Journal of Mechanical Engineering Science, 1998, agg Borghi M., Milani M., Paoluzzi R. - Transient Flow Forces Estimation on the Pilot Stage of a Hydraulic Valve - Proceedings of the 1998 IMECE - ASME International Mechanical Engineering Congress and Exosition - Fluid Power systems Technology Division Anaheim, California, November 15-20, Borghi M., Cantore G., Milani M., Paoluzzi R. - Numerical Analysis of Lateral Forces Acting on Sools of Hydraulic Comonents - Proceedings of the 1998 IMECE - ASME International Mechanical Engineering Congress and Exosition - Fluid Power systems Technology Division Anaheim, California, November 15-20, Borghi M., Cantore G., Milani M., Paoluzzi R. - Hydraulic Locking in Sool-tye Valves Joint reort, University of Modena, CEMOTER C.N.R. 7. J.F. Blackburn, G. Reethof, L. Shearer Fluid Power Control Technology Press of MIT and John Wiley & Sons, H.E. Merrit Hydraulic Control Systems - John Wiley & Sons, Borghi M., Milani M., Paoluzzi R., Zarotti G.L. - Visualizzazione delle Traiettorie del Contaminante Solido in Comonenti Oleodinamici - Atti del 53 Congresso Nazionale ATI Firenze Settembre (in Italian). 10. K.L., Johnson - Contact Mechanics - Cambridge University Press ISBN D.C. SWEENEY: Preliminary Investigation of Hydraulic Lock, Engineering, ,

13 APPENDIX Non-dimensionalization rocedure reminder NOMENCLATURE Symbols L length U, u, w velocity C C A C Q ressure re-scale factor area re-scale factor flow-rate re-scale factor ressure C F force re-scale factor Re Reynolds number β(re) Reynolds number ratio arameter ρ Fluid density K resistance coefficient µ Fluid dynamic viscosity C d discharge coefficient ν Fluid cinematic viscosity D/Dt substantial derivative t σ A Q F C u time Total ressure tensor area flow-rate force velocity re-scale factor Suerscrit Or non-dimensional values Subscrit reference values 0 hysical roerties reference values of fluid st restriction, metering edge NON-DIMENSIONAL APPROACH The non-dimensional statement of any comutational roblem starts from the choice of a set of reference values 1 for length, L, velocity, U, density, ρ 0, and dynamic viscosity, µ 0. This set is intended to reresent the geometric and fluid dynamic asects of the henomena under investigation. The scale factors derived lead to the statement of the following set of non-dimensional variables for the roblem ( denotes dimensionless quantities): L = L L u = u U (1) t U = t L = (2) ρ 2 0 U 1 Normally the choice of this set of values refers to a reresentative section of the fluid flow field: however, no roblems would derive from a non-dimensional statement that refers to a general set of values. 13

14 Keeing in mind this set of arameters, the descritive equations of the roblem can be rewritten and solved in the non-dimensional form. In the case of fluid flow, starting from the general dimensional Navier-Stokes equation, ρ Du = + µ 2 u Dt (3) it is ossible to refer the differential oerators to non-dimensional terms by introducing a coherent dimensional factor, and Equation (3) can be rewritten in the form: Du Dt ρ0 = ρ µ + ρ L 2 u U (4) Equation (4) now contains non-dimensional differential terms only, the reference values for length and velocity and a correlation between actual values, (ρ, µ), and reference values, (ρ 0, µ 0 ), of the hysical roerties of the fluid. In order to obtain unitary non-dimensional values, a convenient choice for Newtonian, incomressible and isothermal flows is to use the actual values as reference quantities. Introducing the non-dimensional terms for density and dynamic viscosity as well, and the reference Reynolds number defined for the roblem (Re ): ρ µ ρ = = 1 µ = = 1 ρ µ 0 0 (5) ρ U L Re = (6) µ the final form of the non-dimensional governing equation becomes: Du Dt = + 1 Re 2 u (7) The structure of Equation (7) is the same as that of Equation (3); differences are limited to the constant coefficient values. This analogy in structure of dimensional and non-dimensional equations allows the use of numerical codes for the solution of both dimensional and non-dimensional equation sets. For instance, Equation (7) becomes identical to Equation (3) if the actual values for density and dynamic viscosity are set as: ρ = 1 µ = 1 Re (8) 14

15 Conversely, rearranging Equation (7) as: Re Du Dt = Re + 2 u (9) it is ossible to write: Re Du Dt = + 2 u (10) where the non-dimensional ressure is now defined as: Re = Re = 2 ρ0 U (11) In erfect analogy with the revious case, Equation (10) turns back into Equation (3) assuming: ρ = Re µ = 1 (12) Equations (7) and (10), although exressing two different non-dimensional statements of the fluid dynamic roblem, reresent the same fundamental equation. Some authors [1] suggest that the use of Equation (10), couled with Equation (12), gives better results for numerical analysis of low Reynolds number flows, while Equation (7) and (8) must be introduced during analysis of high Reynolds number flows. With regard to Newtonian, incomressible and isothermal flows inside hydraulic comonents, the main oints of interest are ressure 2 and velocity distributions: from these basic quantities it is ossible to derive information about other henomena of interest in fluid ower design, such as forces acting on moving arts, ressure dros, flow-rate assing through a section, energy distribution and dissiation, discharge effects, Coanda effect, and so on. The non-dimensional information is normally used to the urose of dimensional characterisation of both the roblem under study and every other similar (in geometry and flow) fluid dynamic roblem. In fact, should comlete similarity exist, the non-dimensional values obtained can be easily scaled using the re-scaling equations related to different, though similar, roblems. The aroach roosed extends dimensional re-scaling to cases where only geometric similarity exists. 2 Here the term ressure is referred to the total ressure tensor, whose generic term is given by: σ ( u u ) i, j = δ i, j + µ i, j + j, i 15

16 Starting from a numerical solution for a given non-dimensional statement, the quantitative information is obtainable only with the conversion of this solution to a dimensional form: at this stage, the roer set of dimensional coefficients to re-scale the roblem must be used, reversing the rocedure reviously described. In detail, the general form of the main re-scaling factors (hereinafter indicated as C) is: C u 1 = ν Re L (13) C ( lowre) 2 2 = ρ ν Re L (14) C ( high Re) 2 2 Re 2 = ρ ν L (15) Using the exressions above it is now ossible to characterise the dimensional roblem comletely, obtaining all the information related to the numerical flow field; the derived quantities can be comuted starting from the basic set of dimensional variables, and the relevant conversion factors can be obtained in erfect analogy with the rocedure used for basic quantities. For examle, the flow-rate through a section or the force alied on a surface can be derived from ressure and velocity distributions: Q = u A = C u C A Q (16) F = A = C C A F (17) where C A = L 2 is the scale factor for areas. The scale factors for velocity and area do not deend on the tye of flow considered, so the evaluation of dimensional flow-rate is unique, while the force comutation deends directly on ressure distribution and is subject to modifications according to the tye of flow assumed. Scale factors for flow-rate and force comutation therefore become: C Q = C C = ν Re u A L (18) C F ( lowre) = C ( lowre) C = ρ ν Re (19) A 2 C ( Re) ( Re) 2 Re 2 F high = C high C A = ν ρ (20) 16

17 EXPANDING A NON-DIMENSIONAL SOLUTION The geometrical comlexity of fluid ower comonents and their normal oerating conditions usually render the flow field turbulent (exerimental evidence suggests transition values of the Reynolds number between 150 and 600, [4]). In view of this behaviour, it is advisable to use the high Reynolds number aroach of Equations (15) and (20) to non-dimensional equation set definition. As to further effects, it must be noted that in the case of relatively high Reynolds numbers for the roblem under study the second term on the right-hand side of Equation (7) becomes negligible, and the non dimensional total ressure tensor can be aroximated by its hydrostatic comonent: σ = + 1 Re u (21) For ractical uroses, both non-dimensional equations (Equations (7) and (21)) can be simlified by directly ignoring the term related to the effect of tangential stress on a generic fluid element [5], and it seems ossible to describe the comlete fluid dynamic roblem by simly alying the generic deendency of the flow-rate on the square root of the differential ressure, considering only the saturated behaviour of the comonent. Consequently, the assumtion of a fully develoed turbulent field imlies that the roblem can be treated as a sudden contraction-exansion situation. In the case of hydraulic sool tye valves, for examle, this means acceting that all fluid dynamic effects occur at, or near, the metering edge defined by the axial osition of the sool. Now, assuming for a generic comonent: a reference non-dimensional geometry; a reference working condition (high Reynolds number); a fully develoed turbulence flow; a Newtonian incomressible fluid; a stationary isothermal analysis, it is ossible to solve the roblem for the given reference non-dimensional working condition in order to obtain the seed distribution and the ressure field over the domain. 17

18 Once a converged non-dimensional solution is obtained, the trile 0 0, 0 Q, F can be comuted and as the reference Reynolds number chosen ( Re ) is known, the scale factors used to obtain the dimensional solution are: 0 C Q0 = ν Re 0 L (22) C 0 ( high Re) = ρ ν Re L 0 (23) C 2 2 F ( high Re) = ρ ν Re 0 0 (24) The terms of these dimensional scale factors are such that, at fixed geometry, the non-dimensional values obtained by the numerical integration are re-scaled according to a transformation that establishes a deendency of the tye: Q in out (25) If the comonent is considered as a concentrated loss, the scale factors above define a transformation, which enables a comlete characterisation of the comonent behaviour by simly starting from the initial nondimensional trile. Obviously, the roortionality between flow-rate and ressure dro will be only a function of the geometry, the fluid density and the discharge coefficient, C d. The influence of the last factor (C d ) is to a large extent unknown; however, for a fully develoed turbulent flow, exerimental evidence shows that the coefficient can be considered constant with resect to the Reynolds number of the flow [4], [7], therefore the extension of the solution comuted for a single Reynolds number to a continuous range seems to be justified (within the validity limits of a fully develoed turbulent flow involving a Newtonian, incomressible fluid). According to Equations (16) and (18), the dimensional solution for a different flow-rate condition can now be determined using a different Reynolds number, Re : 1 Q =ν Re 1 1 L 0 Q (26) Introducing the arameter β ( Re), as the ratio between the Reynolds number for a new flow-rate condition and the reference value chosen for the non-dimensional statement of the numerical roblem: β ( Re) Re = Re 1 0 (27) 18

19 Equation (26) becomes: ( ) 0 Q1 = β Re Q (28) With the same considerations, dimensional relations for ressure and force determination in a new flow-rate condition can be exressed as follows: ( Re) ( Re) 1( high Re) = β 0 high 2 ( Re) F ( Re) F1 ( high Re) = β 0 high 2 (29) (30) The choice of the new flow-rate condition is comletely free within the validity limits of the flow model, therefore the use of this re-scaling technique allows a direct construction of the dimensional characteristic curves commonly used in engineering ractice (Q-, F- and F-Q). 19