1

Using Steady Flow Force for Unstable Valve Design:Modeling and Experiments

Qinghui Yuan and Perry Y. Li

Abstract— In single stage valves, the main spoolsare stroked directly by solenoid actuators. Theyare cheaper and more reliable than multistagevalves. Their use, however, is restricted to lowbandwidth and low flow rate applications due tothe limitation of the solenoid actuators. Our re-search focuses on alleviating the need for large andexpensive solenoids in single stage valves by ad-vantageously using fluid flow forces. For example,in a previous paper, we proposed to improve spoolagility by inducing unstable transient flow forces bythe use of negative damping lengths. In the presentpaper, how steady flow forces can be manipulatedto improve spool agility is examined through fun-damental momentum analysis, CFD analysis andexperimental studies. Particularly, it is found thattwo previously ignored components - viscosity ef-fect and non-orifice momentum flux, have strong in-fluence on steady flow forces. For positive dampinglengths, viscosity increases the steady flow force,whereas for negative damping lengths, viscosity hasthe tendency of reducing steady flow forces. Also,by slightly modifying the non-orifice port geometry,the non-orifice flux can also be manipulated so as toreduce steady flow force. Therefore, both transientand steady flow forces, can also be used to improvethe agility of single stage electrohydraulic valves.

Keywords: Unstable valve, damping length, steadyflow force, viscosity, non-orifice flux, computationalfluid dynamics (CFD)

This research is supported by the National Science Founda-tion ENG/CMS-0088964. Submitted to the ASME Journal ofDynamic Systems, Measurement and Control, July 2003. Por-tions of this work appeared or will appear at the 2002 BathWorkshop on Power Transmission and Motion Control, 2002American Control Conference, and 2003 ASME-IMECE.

Q-H Yuan and P. Y. Li are with the Department of MechanicalEngineering, University of Minnesota, Minneapolis, MN 55455.E-mails: {qhyuan, pli}@me.umn.edu. Please send all corre-spondence to Professor Perry Y. Li.

Nomenclature

α viscosity related geometry coefficient∆1 land area difference compensation∆2 land tapers compensationτsleeve wall stresses on the sleeveτrod wall stresses on the rodθ Vena contracta jet angleµ dynamic viscosityρ densityAl annular area of the left landAo orifice areaAr annular area of the right landCc Vena contracta contraction coefficientcin meter-in chamber non-orifice outlet flux

coefficientcout meter-out chamber non-orifice inlet flux

coefficientFefflux net momentum effluxFland pressure force acting on the landsFrod viscous force acting on the rodFsleeve viscous sleeve force acting on fluidFspool fluid flow force acting on spoolFsteady steady flow forceL damping lengthL1 meter-in chamber damping lengthL2 meter-out chamber damping lengthLd dead space distancen unit vector outward normal to surfacenx positive x unit vectorkL steady force sensitivity to damping

lengthPl pressure on the left landPr pressure on the right landQ flow rateP fluid pressureRi radius of the rodRo radius of the sleevev fluid velocity vectorvx longitudinal fluid velocity

2

I. Introduction

In a single stage electrohydraulic flow control valve,

the main spool is stoked directly by solenoid actua-

tors. In high flow rate and high bandwidth applica-

tions, the solenoids must overcome significant forces.

The force and power limitations of the solenoid ac-

tuators make single stage valves un-suitable for high

performance applications. In these situations, multi

stage valves are generally used in which the spools are

driven by one or more pilot stage hydraulic valves.

However, multi stage valves tend to be more suscep-

tible to contamination, and more expensive in terms

of manufacturing and maintenance, than single stage

valves.

Our research aims at improving the flow and band-

width capabilities of single stage valves by alleviating

the need for large and expensive solenoid actuators.

Our approach is to utilize the inherent fluid flow forces

to increase the agility of the spool, so that less power

or less force is needed from the solenoids while achiev-

ing fast spool responses. Flow forces can be classified

as either steady or transient. Steady flow forces are

those exerted by the fluid on the spool during steady

flow conditions, while transient forces are the addi-

tional forces due to the time variation of the flow con-

dition. Our initial approach [1] proposes to configure

the valve to achieve transient flow force that has an

unstable (negative damping) effect on the spool, and

then to stabilize the system by closed loop feedback.

Analysis and computer simulations show that valves

configured to have unstable transient flow forces can

have faster step responses under solenoid saturation

than their stable counterparts. They also take less

positive power but more negative power (braking)

Ps Sleeve Pt

Fe

Q Q

Rod

Xv

L1 L2

Actuator

Land

Solenoid

Fig. 1. Typical configuration of a four way direction flow

control valve. The two “Q” ports are connected to

the load (hydraulic actuator), and Ps is connected to

the supply pressure, and Pt is connected to the return.

In a single stage valve the spool is stroked directly by

solenoid actuators. Damping length L is defined to be

L := L2 − L1.

power to track a sinusoidal flow rate. The saving in

positive power becomes more significant at higher fre-

quencies. A basic premise of this study is that the

sign of damping length determines whether the tran-

sient flow force is stabilizing or destabilizing (see Fig.

1 for a definition). The study in [1] distinguishes from

other studies of transient flow force induced instabil-

ity in the 1960’s [2], [3] in that our goal is to utilize the

instability advantageously rather than to eliminate it

as in the previous studies.

In this paper, experimental results are first pre-

sented to confirm that spool agility is indeed improved

when valves are configured to have small or nega-

tive damping lengths. Then, it is argued that the

improved agility may not due solely to the unstable

transient flow forces as was originally hypothesized.

The flow force models in [1] as well as in [2], [3] fo-

cus only on the momentum flux at the variable orifice

and neglect the effects of viscosity and the non-orifice

3

momentum flux. They predict that the steady flow

forces should be unaffected by damping lengths. In

this paper, we present models for flow forces that take

into account fluid viscosity and non-orifice momentum

flux. The new models are developed based on funda-

mental momentum theory and are verified using com-

putational fluid dynamics (CFD) analysis and experi-

ments. It turns out that both viscosity and non-orifice

momentum flux have significant effects on steady flow

forces, and hence spool agility. In particular, viscos-

ity reduces the steady flow force for negative damping

lengths, and increases it for positive damping lengths.

For sufficiently negative damping lengths, the steady

flow force can even become unstable. Thus, the exper-

imentally observed improvement in spool agility for

negative damping lengths, can be attributed, to the

reduction in steady flow forces in addition to the un-

stable transient flow force. Furthermore, we show that

the non-orifice momentum can be manipulated by mi-

nor modification of the port geometry, so as to reduce

the steady flow force. Consideration of the viscosity

effect and the non-orifice momentum flux presents ad-

ditional design opportunities to improve spool agility

via the reduction of steady flow forces.

Steady flow force compensation techniques have

been studied in the 1960’s (see [3], [2] for a summary).

These focus on altering the flow pattern at the vari-

able orifice (which involves shaping the lands or the

sleeves, or using multiple small orifices) or on induc-

ing a differential force on the spool ends at high flows.

The steady flow force reduction methods that make

use of viscosity or the non-orifice momentum flux may

be simpler to implement.

The rest of paper is organized as follows. In section

II, we present some experimental results that confirm

the influence of damping lengths on spool agility. In

section III, we present models for the fluid flow in-

duced steady flow forces that take into account vis-

cosity and non-orifice momentum flux. Section IV

presents CFD analysis of the fluid forces and compare

the accuracies of the various approximations. The ex-

perimental study on the viscosity effect and the non-

orifice flux effect is presented in section V. Section VI

contains some concluding remarks.

II. Effect of damping Lengths on Spool

Agility: Experimental results

The experimental setup (Fig. 2) consists of a cus-

tom built valve that allows for different (both pos-

itive and negative) damping lengths L. The valve

sleeve has eleven (11) ports, fitted with quick couplers

for connection to the inlets and outlets. The ports

that are not connected are blocked. By connecting

the supply and return to different ports, the dimen-

sions L1 and L2 in Fig. 1 and the damping length

L := L2 − L1 can be changed. Although both meter-

in and meter-out chambers (see Section III) can be

achieved simultaneously, our experiments investigate

each single chamber separately so that either L1 = 0

or L2 = 0. The spool consists of a threaded hardened

precision anodized aluminum shaft on which several

bronze lands can be arbitrarily positioned to be con-

sistent with the chosen inlet / outlet locations. The

spool is actuated by a pair of solenoids and the spool’s

displacement is measured by a linear potentiometer.

The solenoids are driven by a MOSFET circuit so that

the gate voltage Vg applied to the MOSFET would be

roughly proportional to the solenoid current. The con-

4

(a) Photograph of the valve

A B C D E F G H I J K

6.35 mm

1 2

. 7 m

m

216 mm 117 mm

(b) Dimension of the valve

Fig. 2. Custom built valve with adjustable damping

length. By connecting the inlet / outlet and actua-

tor to the various ports, different damping lengths can

be achieved. The configuration shown has only one

valve chamber.

trol of the solenoid and the data acquisition system

are coordinated by a PC within the Matlab (Math-

works, MA USA) environment via the Matlab’s Real-

time Toolbox (Humusoft, CZ Rep.).

The spool displacement trajectories in response to

step inputs of Vg = 2.3V for the various damping

lengths are shown in Figure 3. As the damping length

decreases, it takes less time for the spool to travel the

full stroke of 7 mm (transit time). The transit times

for Vg = 2.3V and Vg = 2.4V for the different damping

lengths are shown in Fig. 4. In both cases, the transit

times decrease as damping length decreases. These

experimental results show clearly that damping length

has a significant effect on spool agility. Moreover, the

damping length has a more significant effect for Vg =

2.3V than for Vg = 2.4V as indicated by the slopes in

20 30 40 50 60 70 80−2

0

2

4

6

8

10x 10

−3

time(ms)

x v(m)

−0.216

−0.118

0.118

0.216

Fig. 3. Spool displacement trajectories for various damp-

ing lengths (from left to right: damping lengths are

L = 0.216, 0.118,−0.118,−0.216m.) when a step in-

put of Vg = 2.3V is applied.

Fig.4. This latter result was qualitatively predicted by

the numerical analysis in [1] which shows that spool

instability has a greater effect on spool agility when

the solenoid limitation is more severe. In fact when

Vg = 2.2V , the solenoid limitation is so severe that

the spool is halted by friction and is unable to travel

the entire stroke length (figure 5). However, a longer

stroke is attained when the damping length is smaller.

For L = −0.216m, the spool is able to nearly complete

the entire stroke.

III. Modeling of steady flow forces

The qualitative increase in spool agility observed

in the experiments in Section II was originally ex-

plained by the traditional premise [1], [3], [2] that

transient flow forces tend to be destabilizing for neg-

ative damping lengths. In this section, we show that

the improved agility can also be attributed in part

to reduction in steady flow force due to viscosity and

non-orifice flux effects.

A typical four way directional control valve such as

5

−0.25 −0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.2 0.2528

30

32

34

36

38

40

42

44

46

48

damping length(m)

resp

onse

tim

e(m

s)V

g=2.3V

Vg=2.4V

Fig. 4. Spool transit times versus damping length for step

inputs of 2.3V and 2.4V.

0 20 40 60 80 100 120−1

0

1

2

3

4

5

6

7

8x 10

−3

time(ms)

x v(m)

L= 0.216m

L= 0.118m

L= −0.118m

L= −0.216m

Fig. 5. Displacement trajectory of the spool displacement

various damping lengths for step input of Vg = 2.2V

in Figure 1 consists of a meter-in chamber (left) and

a meter-out chamber (right). The fluid flow induced

force on the spool for the meter-in and the meter-out

chambers are analyzed separately. In single chamber

experiments, negative damping lengths correspond to

meter-in chambers (L = −L1) and positive damping

lengths correspond to meter-out chambers (L = L2).

A. Meter-in chamber

Consider first the meter-in chamber (Fig. 6). The

force Fspool (positive to the right) that the spool ex-

L1

x

Xv

P r P l

−τ sleeve

θ

−τ rod

τ rod

Ld

P l P r

Ri

Ro

(x=0)

Fig. 6. Meter-in valve chamber. The lower gray block is

the control volume (C.V.) which includes all the fluid

in the chamber, and is surrounded by the sleeve, rod,

land ends and the inlet/outlet surfaces.

periences from the fluid can be calculated in various

ways. Most fundamentally, it is given by:

Fspool = Fland + Frod

=

∫

Ar

Pr dA−∫

Al

Pl dA+

∫

rodτrod dA

(1)

where Fland is the pressure force acting on the lands,

Frod is the viscous force acting on the spool rod, Pr

and Pl are the pressures on the right and left lands,

Ar and Al are the annulus area of the right and left

lands, and τrod is the shear stress exerted on the rod

by the fluid.

Fspool can also be computed from the reaction of

the force that the spool acts on the fluid. The conser-

vation of longitudinal (left to right) momentum equa-

tion for the control volume (C.V.) that encompasses

the fluid in the meter-in chamber (marked gray in Fig.

6

6) is given by:

− Fspool +

∫

sleeve−τsleeve dA

︸ ︷︷ ︸Fsleeve

=d

dt

∫

C.V.ρvxdV +

∫

C.S.ρvx v · ndA

︸ ︷︷ ︸Fefflux

(2)

where the LHS is the sum of the forces that the spool

and the sleeve act on the fluid, and the RHS is the

sum of the rate of change of the longitudinal momen-

tum and the net momentum flux out of C.V.. Here, ρ

is the fluid density, v and vx are the fluid velocity and

its longitudinal component, and n is the outward unit

normal vector associated with the differential surface

dA. The areas of and the shear stresses at the inlet

and outlet surfaces are assumed to be negligible com-

pared to the sleeve area and the sleeve shear force.

From (2), Fspool can be decomposed into a steady

flow force component and a transient flow force com-

ponent:

Fspool = −Fefflux + Fsleeve︸ ︷︷ ︸steady flow force

− d

dt

∫

C.V.ρvx dV

︸ ︷︷ ︸transient flow force

. (3)

Previous models for steady flow force [2], [3] do not

consider the viscosity effect, i.e. Frod = Fsleeve = 0.

Consider now Fefflux and Fsleeve separately. The net

momentum efflux in (2) is given by:

Fefflux,1 =

∫

outletρvxv · ndA+

∫

inletρvxv · ndA (4)

since, if there is no leak, the only surfaces that have

non-zero flux through them are the inlet and outlet.

The efflux at the non-orifice outlet is often neglected

in previous studies [1], [2], [3] on the assumption that

the fluid velocity is nearly normal to the spool axis

at the outlet surface (vx = 0). However, CFD analy-

sis in Section IV reveals that the outlet efflux can be

significant.

Generally, knowledge of the flow profile at the inlet

and outlet is needed to evaluate Eq. (4). However,

the flow at the variable orifice inlet is often modeled

by a Vena contracta in which the flow is uniform [3].

If we further assume that the flow profile at the non-

orifice outlet is also proportional to the flowrate Q,

then, Eq. (4) can be approximated by:

Fefflux,2 =

(−cin +

ρ cos θ

CcAo(xv)

)Q2 (5)

where Ao(xv) is the orifice area when the spool dis-

placement is xv, θ is the Vena contracta jet angle,

Cc < 1 is the contraction coefficient so that CcAo(xv)

is the jet area, and cin is a coefficient that summarizes

the normalized flow profile at the non-orifice outlet.

The sign convention in (5) has been chosen so that

cin > 0 in typical port geometries.

Consider now the viscous sleeve force Fsleeve. If the

fluid is Newtonian, then fundamentally,

Fsleeve,1 = −∫

sleeveτsleeve dA =

∫

sleeveµ∂vx∂n

dA (6)

where µ is the fluid dynamic viscosity and ∂vx∂n is the

longitudinal velocity gradient with respect to the out-

ward normal at the sleeve surface. To evaluate Eq.

(6) generally requires knowledge of the flow profile at

the non-orifice outlet.

Using the assumptions that 1) the friction force in

the dead space between the outlet and the left hand

land in Fig. 6 is negligible; 2) the flow is laminar and

fully developed along the full length L1 between the

inlet and outlet; Eq.(6) can be approximated by

Fsleeve,2 = αµL1Q (7)

for some coefficient α > 0 that depends only on valve

7

geometry. Generally, α increases as the cross sectional

area of the flow passage decreases.

For the cylindrical valve geometry in Fig. 6 (see the

Appendix),

α =4(2R2

o ln(Ro/Ri) +R2i −R2

o)

(R4o −R4

i ) ln(Ro/Ri)− (R2i −R2

o)2. (8)

The assumption that the flow in the chamber is lam-

inar is valid when the Reynolds number Re < 2100

(e.g. when the flow rate is sufficiently low). For

our experimental system in Fig. 2, Re ≈ 1000 when

Q = 38 lpm (10 gpm). For higher flow rate, the flow

may become turbulent and the shear stress must in-

clude also the Reynolds stresses caused by the random

velocity components. The analysis can be extended to

this case using friction / flow relation obtained from

tabulated empirical friction factors. For example, in

the fully turbulent regime, the friction factor is pro-

portional to 1/R0.25e [4] and Eq. (7) can be replaced

by

Fsleeve,3 = βµ0.25Q1.75L1 (9)

where β is a function of the geometry and the fluid

density. In this paper, we focus on laminar flows.

In summary, the steady flow force for the meter-in

chamber is given by

Fsteady = −Fefflux + Fsleeve (10)

where Fefflux is given either fundamentally by (4) or

approximately by (5); and Fsleeve is given either fun-

damentally from (6) or approximately by (7).

Since the flow rate Q increases monotonically with

the orifice area Ao(xv), and Ao(xv) is roughly linear

w.r.t. xv, Eqs. (5) shows that the steady flow force

component due to the variable orifice will be roughly

linear in xv. This is the well known stable linear spring

L2

x

Xv θ

P r P l

−τ sleeve −τ rod

τ rod

Ri

Ro

P l P r (x=0)

Ld

Fig. 7. Meter-out valve chamber.

effect on the spool [2], [3]. On the other hand, both

the non-orifice flux component (−cinQ2) in Eq. (5)

and Fsleeve in Eq.(7) correspond to unstable quadratic

and unstable linear spring forces with negative spring

constants. Therefore, for the meter-in chamber, both

the previously neglected viscosity effect and the non-

orifice flux effect tend to reduce the steady state flow

force.

B. Meter-out valve chamber

A similar analysis can be performed for the meter-

out valve chamber (Fig. 7). The steady flow force is

also given by Eq. (10):

Fsteady = −Fefflux + Fsleeve (10)

with Fefflux given by one of the following:

Fefflux,1 =

∫

outletρvxv · ndA+

∫

inletρvxv · ndA (11)

Fefflux,2 =

(−cout +

ρ cos θ

CdAo(xv)

)Q2 (12)

where cout is a coefficient that summarizes the flow

pattern at the non-orifice inlet for meter-out cham-

ber. The sleeve force is given by one of the following

8

expressions:

Fsleeve,1 = −∫

sleeveτsleeve dA =

∫

sleeveµ∂vx∂n

dA

(13)

Fsleeve,2 = −αµL2Q (14)

where L2 is the distance between the entry port and

the meter-out orifice in Fig. 7. For the valve geometry

in Fig. 7, the same α given in Eq. (8) for the meter-in

case can be used.

For the meter-out chamber, the steady flow force

due to the momentum flux at the variable orifice can

again be represented by a stable linear spring. The

non-orifice momentum flux (−coutQ2) also acts like

an unstable quadratic spring with a negative spring

constant and serves to reduce the steady flow force.

Unlike the meter-in chamber, the sleeve force compo-

nent of the steady flow force in Eq.(14) now acts like a

stable spring with a positive spring coefficient. There-

fore, the neglected viscosity effect tends to increase

the steady flow force in the case of a meter-out cham-

ber. Moreover, since typically, 0 < cout < cin, the

reduction in steady flow force due to the non-orifice

flux in a meter-out chamber (L = L2) will be less than

that in a meter-in chamber (L = −L1). Both of these

effects can contribute to the observed improvement in

spool agility in Section II.

C. Valve with both meter-in and meter-out valve

chambers

In a 4-way symmetric valve, the spool is acted on

by the flow forces in both the meter-in and the meter-

out chambers (Fig. 1). Hence, the net force that

acts on the spool is the sum of the forces and can be

approximated by:

Fsteady =

[−(cin + cout) +

2ρ cos θ

CcAo(xv)

]Q2

− αµ (L2 − L1)︸ ︷︷ ︸L

Q (15)

where it is assumed that for a given spool displace-

ment, the Vena contracta coefficient is the same for

both meter-in and meter-out chambers in a symmetric

valve. In commercial valves, L := L2−L1 is designed

to be positive to ensure positive damping effect. In

[1] we proposed that by choosing L < 0 to induce

a negative damping effect through the transient flow

forces so as to improve the agility and responsiveness

of the spool. The new results in this paper are that,

the steady flow forces can also be used to improve the

agility of the single stage valve via the viscosity effect

and the non-orifice flux.

IV. CFD analysis of flow forces

In this section, we present CFD analysis to ver-

ify and evaluate the various flow force models pre-

sented in Section III. The 3D computational mod-

els for a given xv are shown in Fig. 8. Notice that

the valve is not axis symmetric because of the inlet

and outlet ports. The mesh and the boundary condi-

tion for each geometry are generated by the GAMBIT

pre-processor. Each computation volume uses about

1,000,000 nodes and 500,000 elements.

The incompressible Navier-stokes equations with-

out body forces are given by [5]:

Continuity:

∇ · v = 0 (16)

Momentum:

ρ∂v

∂t+ ρv · ∇v = −∇P + µ∇2v (17)

9

(a) L1 = 0.216m or L2 = 0.216m

(a) L1 = 0.118m or L2 = 0.118m

Fig. 8. Two fluid models for a given xv. The orifice is

at the right hand side port. To model the meter-in

chamber (left chamber in Fig. 1), the right hand side

port is the entry port, and the left hand port is the

outlet port. To model the meter-out chamber (right

chamber in Fig. 1), the left hand side port is the entry

port, and the right hand side port is the outlet port.

where ρ is the fluid density, v is the fluid velocity

vector, P is the pressure and µ is the dynamic vis-

cosity. The SIMPLE pressure correction approach

[5] is applied to decouple the continuity and mo-

mentum equations. No slip conditions are imposed

on all land faces, rod and sleeve walls. Fluid den-

sity of ρ = 871kg/m3, and dynamic viscosity µ =

0.0375kg/m/s are used. These correspond to a typ-

ical hydraulic fluid (Mobil DTE 25) at 40◦C. The

rod and inner sleeve radii of the valve are Ri =

Fig. 9. Flow patterns and estimation of jet angles

for the meter-in case: (from left to right) xv =

−0.635,−1.27,−1.905 and −2.54 mm.

3.175mm and Ro = 6.35mm. Sixteen models corre-

sponding to combinations of four spool displacements

xv = 0.635mm,1.27mm, 1.905mm and 2.54mm, and

four single chamber damping lengths L = −L1 =

−0.216m, −0.118m (meter-in chamber) and L = L2 =

0.118m, 0.216m (meter-out chamber) are investigated.

Constant inlet pressure Ps = 689475.7Pa (100psi) and

outlet pressure PT = 101300Pa (1 atm) are imposed.

Solutions are obtained using FLUENT 6 (Fluent Inc.,

NH) on the IBM SP supercomputer at the University

of Minnesota. The various components of the steady

flow force and momentum fluxes can be directly mea-

sured and are used to evaluate the various models in

Section III.

A. Variable Orifice Flux

First, we investigate the variable orifice flux. The

meter-in and meter-out flow patterns are shown in

Figs. 9-10. The key parameter group cos(θ)/Cc in

Eq.(5) is potentially a function of the spool displace-

ment. The jet angle θ for both meter-in and meter-out

chambers at different spool displacements may be ob-

tained from the flow patterns shown in figures 9 and

10. For sufficiently large xv, θ ≈ 69◦ as predicted in

[3]. A contraction coefficient of Cc = 0.6 has been

cited in the literature [3]. This gives an estimate of

cos(θ)/Cc ≈ 0.60.

10

Fig. 10. Flow patterns and estimation of jet an-

gles for meter-out case (from left to right) xv =

0.635,1.27,1.905 and 2.54 mm.

0 1 2 3 4 5 6 70

0.5

1

1.5

2

2.5

3

3.5

ρ Q2/ Ao (N)

Var

iabl

e or

ifice

flux

(N)

L=−0.216mL=−0.118mL=0.118mL=0.216m

Fig. 11. CFD computed variable orifice momentum versus

ρQ2/Ao(xv). The solid line is the estimate based on

the least squares fit with cos(θ)/Cc = 0.454.

Fig. 11 shows the relationship between the

measured variable orifice momentum flux and

ρQ2/Ao(xv), where Ao(xv) is the orifice area of the

circular ports used in this study. The linearity of the

data indicates that a constant least squares estimated

average cosθ/Cc = 0.454 suffices for all spool displace-

ments and damping lengths. This value suggests that

an equivalent contraction coefficient of Cc ≈ 0.79,

which is larger than suggested in [3].

B. Non-orifice Flux

Next, we consider the non-orifice flux component of

the steady flow force. To illustrate, the flow patterns

at the non-orifice inlet / outlet for a 2D (i.e. 3D ax-

−0.2 −0.1 0 0.1 0.2 0.30

0.5

1

1.5

2

2.5

3

Damping length (m)

Var

iabl

e O

rific

e flu

x (N

)

xv=0.635mmxv=1.27mmxv=1.905mmxv=2.54mm

Fig. 12. Variable orifice fluxes as the function of L, for

various xv. The red solid lines are direct CFD mea-

surements, while the blue dotted lines are estimated

based on cos(θ)Cc

= 0.454.

(a) meter-out (inlet) (b) meter-in (outlet)

Fig. 13. Non-orifice flow pattern with the hose designed

to be normal to the spool axis.

ial symmetric) valve model are shown in Figure 13.

Notice that despite the hoses are normal to the spool

axis, the flow patterns at the non-orifice inlet or out-

let are not normal to it, so that non-zero longitudinal

momentum fluxes are expected.

Figure 14 shows the relationship between the mea-

sured non-orifice flux and Q2 for various flowrates

and damping lengths. The linearity of the meter-

in and meter-out data confirms that the momentum

fluxes in Eqs. (5) and (12) can indeed be mod-

eled using constants cin = 4.26 × 106Ns2m−6 and

cout = 1.79× 106Ns2m−6 for all flows, spool displace-

11

0 0.2 0.4 0.6 0.8 1

x 10−7

−0.45

−0.4

−0.35

−0.3

−0.25

−0.2

−0.15

−0.1

−0.05

0

Q2 (m6 s−2)

Non

orif

ice

flux

(N)

meter−in (L=−0.216m)meter−in (L=−0.118m)meter−out (L=0.118m)meter−out (L=0.216m)

Fig. 14. Non-orifice flux as a function of Q2 for the sixteen

3D CFD models. The regression lines correspond to

cin = 4.26×106Ns2m−6 and cout = 1.79×106Ns2m−6

ments, and damping lengths. Notice that the non-

orifice outlet flux coefficient cin in the meter-in cham-

ber is more than twice that of the non-orifice inlet

flux coefficient cout in the meter-out chamber. This

is consistent with the observation in Fig. 13 that the

flow is less obtuse in the meter-in outlet than in the

meter-out inlet.

To further confirm that cin and cout are not de-

pendent on damping length and spool displacement,

extensive studies using 2D (equivalent to 3D axis-

symmetric) CFD models with multiple flowrates at

each spool displacements and damping lengths were

performed. Figure 15 shows that cin and cout can be

treated as constants in these 2D cases as well.

C. Viscosity effect

Fig. 16 shows Fsleeve, computed from the funda-

mental equations (6) and (13) as a function of µQL for

the 16 test geometries. The linear nature of the curve

indicates that the laminar and fully developed flow ap-

0 0.5 1 1.5 2 2.5 30.5

1

1.5

2

2.5

3x 10

5

xv (mm)

cin

an

d c

out (N

s2/m

6)

cin

c

out

−250 −200 −150 −100 −50 0 50 100 150 200 2500.5

1

1.5

2

2.5

3x 10

5

damping length (mm)c

out (N

s2/m

6)

cin

c

out

Fig. 15. Estimated cin and cout for various spool displace-

ments and damping lengths L in a 2D CFD model.

Top: Various xv and L = 0.15m. Bottom: Various L

and xv = 0.645mm.

proximations in (7) and (14) are accurate. The least

squares estimate for the coefficient α is 1 × 106m−2

and the one computed analytically from Eq. (8) is

0.8× 106m−2. Fig. 16 shows that these estimates are

quite close.

D. Total Steady Flow Force

Consider the following expressions for the steady

flow force that use different approximations:

Fsteady,0 = Fland + Frod (18)

Fsteady,1 = −Fefflux,1 + Fsleeve,1 (19)

Fsteady,2 = −Fefflux,2 + Fsleeve,2 (20)

Fsteady,inviscid = −Fefflux,2 (21)

12

−3 −2 −1 0 1 2 3

x 10−6

−4

−3

−2

−1

0

1

2

3

4

µQL(Kg m3 /s2)

Sle

eve

forc

e (N

)

Fsleeve,1

Fsleeve,2

Fig. 16. Measured sleeve force Fsleeve as the function

of µQL. Sixteen diamond points correspond to the

sleeve forces computed from sixteen CFD models, and

the dotted line is the least squares fit. The solid line is

calculated from Eq. (7) or Eq. (14) with α computed

from Eq. (8).

where Fsteady,0 is the most fundamental computed di-

rectly based on Eq.(1); Fsteady,1 is the approach based

on the fundamental momentum equation in Eqs. (4),

(6) and (13); Fsteady,2 is based on the approximate mo-

mentum equations Eqs. (5), (7), (12), (14) using the

estimated parameters; Fsteady,inviscid is the estimate of

the steady flow forces that neglects viscosity. Notice

in Fig. 17 shows that Fsteady,0, Fsteady,1, and Fsteady,2

are almost exactly the same. Both the fundamental

and approximate momentum methods to calculate the

steady flow forces accurate. Furthermore, Fig. 17 and

18 shows that the viscosity effect plays a significant

role in determining the steady flow forces. It reduces

the steady flow force for negative damping lengths,

while increasing it for positive damping lengths. The

extent to which the steady flow force varies increases

with the damping length.

(a)L = −L1 = −0.216m

0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4

x 10−3

−6

−5

−4

−3

−2

−1

0

xv (m)

Stea

dy flo

w fo

rce

(N)

Fsteady,0

Fsteady,1

Fsteady,2

Fsteady,Inviscid

(b)L = −L1 = −0.118m

0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4

x 10−3

−6

−5

−4

−3

−2

−1

0

xv (m)

Stea

dy flo

w fo

rce

(N)

Fsteady,0

Fsteady,1

Fsteady,2

Fsteady,Inviscid

(c)L = L2 = 0.118m

0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4

x 10−3

−6

−5

−4

−3

−2

−1

0

xv (m)

Stea

dy flo

w fo

rce

(N)

Fsteady,0

Fsteady,1

Fsteady,2

Fsteady,Inviscid

(d)L = L2 = 0.216m

0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4

x 10−3

−6

−5

−4

−3

−2

−1

0

xv (m)

Stea

dy flo

w fo

rce

(N)

Fsteady,0

Fsteady,1

Fsteady,2

Fsteady,Inviscid

Fig. 17. Steady flow forces computed from various meth-

ods as a function of the orifice displacement. Four

different damping lengths are considered.

13

−0.2 −0.1 0 0.1 0.2 0.3−6

−5

−4

−3

−2

−1

0

1

Damping length (m)

Ste

ady

forc

e (N

)

xv=0.635mmxv=1.27mmxv=1.905mmxv=2.54mm

Fig. 18. Steady flow forces as a function of the single

chamber damping length.

Fig. 18 shows the dependence of the steady flow

force for each meter-in / meter-out chamber on damp-

ing lengths. Notice the discontinuity of the flow

force at L = 0. This is related to the difference

between the meter-in and meter-out non-orifice flux,

(cin − cout)Q2 so that the discontinuity decreases for

smaller xv (hence Q). Fig. 19 shows that total steady

flow force for the two-chamber 4 way valve according

to Eq. (15). Compared to Fig. 18, no discontinuity

is present because both the non-orifice inlet and out-

let fluxes are simultaneously present in all damping

lengths. Fig. 19 also shows that both the viscos-

ity and non-orifice flux are more significant at high

flow rate. The non-orifice effect is illustrated by the

parallel shift and is given by (cin + cout)Q2. CFD

analysis reveals that the non-orifice flux as a propor-

tion of the variable orifice flux will be even larger for

axis-symmetric (i.e. 2D) valves, than in the 3D axial-

assymmetric case in Fig. 8.

−0.2 −0.1 0 0.1 0.2 0.3−10

−9

−8

−7

−6

−5

−4

−3

−2

−1

0

A

A

A

A

B

B

B

B

Damping length (m)

Ste

ady

forc

e (N

)

xv=0.635mmxv=1.27mmxv=1.905mmxv=2.54mm

Fig. 19. Steady flow forces and their estimates in a two

chamber 4-way valve as a function of the damping

length for fours sets of spool displacements. “A” -

take into account non-orifice flux, “B” - do not take

into account non-orifice flux. The horizontal bars at

L = 0 indicate the estimates if viscosity effect is ig-

nored.

(a) Meter-out (inlet) (b) Meter-in (outlet)

Fig. 20. Non-orifice flow pattern for a 2D CFD model with

the hose is designed to be +30◦ clockwise rotated from

the normal direction to the spool axis.

(a) meter-out (inlet) (b) meter-in (inlet)

Fig. 21. Non-orifice flow pattern for a 2D CFD model with

with the hose is designed to be −30◦ clockwise rotated

from the normal direction to the spool axis.

14

Hose angle meter-in meter-out sum

30◦ 1.17 0.09 1.26

0◦ 1.00 0.45 1.45

−30◦ 1.20 0.80 2.00

TABLE I

Meter-in and meter-out non-orifice flux for

various non-orifice hoses angles, normalized by

that of the meter-in, 0circ hose angle case.

E. Modifying the non-orifice flux

Since the non-orifice fluxes tend to reduce steady

flow forces, it would be advantageous if it can be ma-

nipulated by design. One straightforward idea is to

change the angle of the non-orifice hose to the spool

axis. Figs. 13, 20 and 21 illustrate the differences in

flow patterns for three hose angles for a 2D (or 3D

axis-symmetric) CFD model. Table I shows the non-

orifice flux values for various hose angles, normalized

by the value for the 0◦ hose angle, meter-in chamber

case. It can be seen that the non-orifice flux is suscep-

tible to changes in geometric configuration of the non-

orifice hose. In particular, by rotating the inlet and

outlet hoses in the counter-clockwise direction, the

non-orifice fluxes in both the meter-in and meter-out

chambers increase. If the hose is rotated 30◦ counter-

clockwise, the sum of the non-orifice fluxes of the two

chambers increases by 37%, thus having the effect of

further reducing the steady flow force.

V. Experimental study

The experiment aims at verifying the steady flow

force models in section III and that the viscosity

and the non-orifice flux have significant effects on the

steady flow forces, and hence on the spool agility.

Inlet/outlet hoses

Blocked quick coupler

opened quick coupler

Orifice

Force sensor

A B C D E F G H I J K

Fig. 22. The diagram of experimental setup for measuring

the steady flow forces

The experimental set up (Fig. 22) uses the custom

built valve in Fig. 2. The geometry of the valve is

similar to the one used in the CFD studies in Section

IV. In order to measure the steady flow force, the

shaft of the valve is connected in series with a force

sensor (67M25A, JR3 Inc.) that can measure both

compression and tension along the spool axis. A dig-

ital flow meter (DFS-3, Digiflow Systems) connected

downstream to the valve and an adjustable bypass

needle valve are used to adjust the desired flow rate

through the valve.

In the experiment, the orifice area is set to be

6×10−6m2. The flow forces were measured in 24 con-

ditions: combination of four positive (L = L2), four

negative (L = −L1) single chamber damping lengths:

L = ±236mm, ±175mm, ±118mm, ±62mm); and

three flow rates Q = 3.7 × 10−5m3/s (2.2LPM),

6.7× 10−5m3/s (4LPM), and 1× 10−4m3/s (6LPM).

Fig. 23 shows the measured steady flow forces and

the corresponding estimates based on Eqs. (15) using

parameters estimated in Section IV. The measured

steady flow force values are qualitatively similar to the

model in Fig. 18 in that i) the steady flow force has

an affine dependence on the damping length; ii) the

15

−250 −200 −150 −100 −50 0 50 100 150 200 250−2

−1

0

Ste

ad

y f

orc

e(N

)

−250 −200 −150 −100 −50 0 50 100 150 200 250−2

−1

0

Ste

ad

y f

orc

e(N

)

−250 −200 −150 −100 −50 0 50 100 150 200 250−2

−1

0

Ste

ad

y f

orc

e(N

)

Damping length(mm)

Fig. 23. Measurement and estimation of steady flow forces

as a function of the damping length, for various flow

rates. The solid line is a regression curves assuming

that for each Q, the sensitivity to damping length kL

is the same for all damping lengths. Top: Q = 2.2

LPM, Middle: Q = 4.0 LPM, Bottom: Q = 6.0 LPM.

sensitivity to damping length kL := ∂Fsteady/∂L in-

creases with flow rate; iii) the steady flow force is dis-

continuous at L = 0 with the discontinuity increases

with flow rate. Notice in particular, that as predicted,

the steady flow force is reduced for negative damping

lengths, and for sufficiently negative damping lengths,

the steady flow force becomes marginally stable or

even unstable (reverses sign).

Notice that for each Q, the steady flow force can

be fitted by two straight lines (L > 0 and L < 0)

using the the same slope kL := ∂Fsteady/∂L. The re-

0 0.5 1 1.5 2 2.5 3 3.5 4

x 10−6

0

0.5

1

1.5

2

2.5

3

3.5

4

µ Q(kg m2 s−2)

k L(N m

−1 )

Fig. 24. Relationship between flow force sensitivity to

damping length kL and flow rate Q. The slope should

be αµ.

lationship between kL and Q are shown in Fig. 24 and

indicates a straight line with a slope which should be

α/µ where µ is the dynamic viscosity. This gives a

value of α = 0.978 × 106 which is close to the val-

ues obtained by CFD in Section IV (1.0× 106) or via

Eq.(8) (0.8× 106).

Despite excellent match in the viscosity effect, there

are some significant quantitative discrepancies be-

tween the experimental results in Fig. 23 and the

model. First, the discontinuity at L = 0 between

the meter-in and meter-out chamber is significantly

larger than that predicted by the expected non-orifice

flux differences. Second, the experimental data tend

to be more negative than the prediction, especially

when L > 0. Extensive effort was spent to investi-

gate the source of these disparities. Two limitations

of our experimental setup can be used for their expla-

nations. Firstly, since the valve lands are machined

by hand, it turns out that the areas of the two spool

lands are different. The area difference was estimated

by measuring the spool force when the closed cham-

16

ber formed using the two lands is pressurized. The

land at the variable orifice was found to be smaller

by ∆A = 0.186mm2. This leads to an extra orifice-

closing force ∆1 ≈ ∆A · P (negative force in the fig-

ure) as the chamber is pressurized. Using the pressure

measured using the digital pressure transducers dur-

ing the experiments, the steady force offset ∆1 are

shown in table II.

Secondly, the shape of the lands need to be consid-

ered. To avoid hydraulic locking and to reduce friction

between the lands and the sleeve, the lands were in-

tentionally machined to have a 2.5◦ taper to form a

hydrostatic bearing. The effect of the taper can be

seen in Fig. 25 where the control volume is different

from that in Figs. 6 and 7. Modifying Eqs. (1)-(2)

accordingly, the spool force of the tapered land valve

is:

Fsteady,t = −Fefflux + Fsleeve + ∆1

+

∫

s−lP (nx · n)dA−

∫

o−pP (nx · n)dA

︸ ︷︷ ︸∆2

(22)

where n is the outward normal vector of the surface,

and nx represents the unit vector in the positive x

direction. The offset force due to the taper, ∆2, con-

sists of the longitudinal pressure force applied on the

orifice plane (o-p) and on the side surface (s-l) respec-

tively. It can be seen that ∆2 > 0 for the meter-in

case because the first term of ∆2 is greater than the

second term, while ∆2 < 0 for the meter-out case

because of the opposite effect. The numerical values

of ∆2 was obtained via CFD analysis of a 2D (i.e.

axis-symmetric 3D) model at various pressure differ-

ences across the orifice. The ∆2 values for the 3D

axis-asymmetric case were then estimated from the

Q (LPM) 2.2 4.0 6.0

Meter-in ∆1 (N) -0.16 -0.20 -0.25

(L < 0) ∆2 (N) 0.17 0.27 0.40

Meter-out ∆1 (N) -0.14 -0.16 -0.20

(L > 0) ∆2 (N) -0.058 -0.089 -0.141

TABLE II

Estimation of ∆1 and ∆2 (N)

x

Side sufrace of the right land

Orifice plane

Fig. 25. Modified control volume of the valve with tapered

lands. The variable orifice land area is also smaller

than the area of the left hand side land. The control

volume is the gray colored block.

2D case with the same pressure difference, corrected

for the port area difference (3D case value is approxi-

mately 50% of the equivalent 2D case value). The spe-

cific values for the experimental conditions are listed

in Table II. The predicted flow forces after incorpo-

rating the offsets ∆1 + ∆2 match the experimental

results very closely (Fig. 26). Even better match can

be obtained by slightly adjusting ∆2. The fact that

∆1 and ∆2 are of different signs for L < 0 explains the

observation that there is less discrepancies for L < 0

than for L > 0 in Fig. 23. In a commercial valve,

where the lands can be precisely machined, the land

area difference is expected to be much smaller, and no

taper or a much smaller taper will generally be used.

Therefore, in practice, the ∆1 +∆2 correction will not

be needed.

17

−250 −200 −150 −100 −50 0 50 100 150 200 250−2

−1

0

−250 −200 −150 −100 −50 0 50 100 150 200 250−2

−1

0

−250 −200 −150 −100 −50 0 50 100 150 200 250−2

−1

0

Damping length L (mm)

Ste

ad

y flo

w fo

rce

s(N

)

Fig. 26. Experimentally measured steady flow force com-

pared with models that take into account limitation of

the experimental setup. Top: Q = 2.2 LPM, Middle:

Q = 4.0 LPM, Bottom: Q = 6.0 LPM.

VI. Conclusion

In this paper, we demonstrate that the fluid viscos-

ity and the non-orifice flux are very important in the

estimation of the steady flow forces. On one hand, in

the negative damping length region, the steady flow

forces, conventionally regarded to be always stabiliz-

ing, can be reduced to be marginally stable or unsta-

ble, thus improving the agility of the spool. Paradoxi-

cally, using a valve with negative damping lengths, the

spool’s agility will improve as the fluid becomes more

viscous. On the other hand, the non-orifice flux can

be tuned by changing the angle of the non-orifice hose

to the spool axis, so that the net momentum efflux,

which is traditionally regarded to be always stabiliz-

ing, could possibly be reduced to be marginally stable

or unstable. The viscosity and the non-orifice effects

predicted by the proposed models and CFD analysis

are verified in experiment.

The viscosity and non-orifice flux effects generate

other valve design parameters that can be used to

improve spool agility in single stage electrohydraulic

valves. Since the steady flow force is a significant force

that the solenoid actuator has to overcome, the reduc-

tion of the steady flow force via the viscosity effect will

be useful for developing high performance single stage

valves.

References

[1] Kailash Krishnaswamy and Perry Y. Li, “On using unsta-

ble electrolydraulic valves for control,” in 2000 ACC Pro-

ceedings, May 2000, Also to appear in ASME Journal of

Dynamic Systems, Measurement and Control.

[2] J. F. Blackburn, G. Reethof, and L. L. Shearer, Fluid Power

Control, MIT Press, 1960.

[3] Hebert E Merritt, Hydraulic Control System, John Wiley

and Sons, 1967.

[4] B. R. Munson, D. F. Young, and T. H. Okiishi, Fundamen-

tals of Fluid Mechanics, John Wiley and Sons, New York,

1998.

[5] John C. Tannehill, Dale A. Anderson, and Richard H.

Pletcher, Computational fluid mechanics and heat transfer,

Taylar and Francis, Philadephia, PA, 1997.

Appendix

For the valve with geometry in Fig. 6, the sleeve

force is fundamentally given by:

Fsleeve,1 = 2πRoµ

∫ 0

−(Ld+L1)

∂vx∂r

∣∣∣∣r=Ro

dx (23)

where ∂vx∂r

∣∣r=Ro

is the longitudinal velocity gradient

in the outward radial (r) direction evaluated at the

18

sleeve wall (r = Ro). If the friction in the dead space

is negligible,

Fsleeve,1 ≈ 2πRoµ

∫ 0

−L1

∂vx∂r

∣∣∣∣r=Ro

dx. (24)

For laminar and fully developed flow, it can be shown

that [4]:1

r

d

dr(rdvxdr

) =1

µ

dP

dx(25)

where P (x) is the pressure. Applying the no-slip con-

ditions that vx = 0 at the rod wall (r = Ri) and the

sleeve wall (r = Ro), the velocity distribution can be

determined to be:

vx =1

4µ

dP

dx

[r2 −R2

o +R2i −R2

o

ln(Ro/Ri)ln

r

Ro

](26)

Since

−∫ Ro

Ri

2πvxdr = Q,

∂vx∂r can be expressed as a function of Q so that α in

(7) is given by:

α =4(2R2

o ln(Ro/Ri) +R2i −R2

o)

(R4o −R4

i ) ln(Ro/Ri)− (R2i −R2

o)2

(8)