hydraulic jumps due to oblique impingement of circular liquid jets on a flat horizontal surface

7/22/2019 Hydraulic jumps due to oblique impingement of circular liquid jets on a flat horizontal surface

1/17


2/17

248 R. P. Kate, P. K. Das and S. Chakraborty

(a )(ii)

H

R j

2r 0

V hext

(a )(i) (b)(i)

R j ()

H

V

= 0 =

Wall jet region

S2S1 S3 S4 S5

r 0(c)

Impingement

n

(b)(ii)

region

Figure 1. Illustrative sketches for ( a)(i) Normal impinging jet, and ( b)(i) oblique impinging jet. (a )(ii) The circular hydraulic jump (circular liquid jet impinging normal to the horizontalsurface), volume ow rate of water, Q = 1 10 4 m3 s 1, nozzle diameter, d = 8 .8mm. (b)(ii)Non-circular hydraulic jump due to a circular liquid jet impinging obliquely on a horizontalsurface, Q = 1 10 4 m3 s 1, d = 8 .8 mm, nozzle inclination (with horizontal), n = / 4.

(c) Regions of ow eld in the case of a normal impinging jet.

respectively (for details see: Watson 1964; Rao & Arakeri 1998). Various aspects of the circular hydraulic jumps, formed under identical conditions, have been extensivelystudied (Watson 1964; Olsson & Turkdogan 1966; Ishigai et al. 1977; Nakoryakov,Pokusaev & Troyan 1978; Craik et al. 1981; Bohr, Dimon & Putkaradze 1993;Godwin 1993; Liu & Lienhard 1993; Higuera 1994, 1997; Blackford 1996; Hansenet al. 1997; Brechet & N eda 1999; and Bush & Aristoff 2003). Oblique impinging jets, in comparison to normal impinging jets, however, have received relatively lessattention in the uid mechanics literature. Figure 1( b)(i) schematically describes an

oblique impinging jet and the associated hydraulic jump characteristics, with thecorresponding photographic view depicted in gure 1( b)(ii). The obliquity of the jetturns out to be a major parameter in determining the nature of the ow in the caseof such non-circular hydraulic jumps, primarily in the following respects. First, the


3/17

Hydraulic jumps due to oblique impingement of circular liquid jets 249

axial symmetry exists only up to the free jet region, and the ow, in general, is three-dimensional elsewhere (Beltos 1976; Rubel 1981). Secondly, the impingement region,which is considered to span over the cross-section of the jet for a normal impinging jet, is expected to change in size and shape, as the jet becomes obliquely inclined to thehorizontal surface. Further, the stagnation point in the impingement zone is no morecoincident with the geometrical centre of the jet. An upstream shift of the stagnationpoint from the geometrical centre of the non-circular impingement zone for obliqueimpinging jets has been reported in the studies of Beltos (1976), Sparrow & Lovell(1980), Stevens & Webb (1991), Rubel (1981, 1982) and Tong (2003). The radial oworiginating from this eccentric stagnation point leads to non-circular hydraulic jumpproles, which is a fundamentally complicated uid dynamic phenomenon. To thebest of our knowledge, even elementary theories for quantitative characterization of the spatial patterns formed as a consequence of such non-circular hydraulic jumps,supported by requisite validation experiments, are yet to be reported in the literature.The aim of the present work, accordingly, is to develop a fundamental understanding

of the geometrical features of non-circular hydraulic jumps, formed as a consequenceof oblique impingement of circular liquid jets from both experimental and theoreticalperspectives, with a quantitative characterization of the pertinent spatial patternsformed.

The outline of this paper is as follows. In 2, laboratory experiments for the circularand non-circular hydraulic jumps will be discussed. In 3, a theoretical analysis forthe location of the stagnation point will be elaborated. Prediction of radial locationsfor non-circular hydraulic jumps will subsequently be made, and these results will befurther compared with the data obtained from experiments. Hydraulic jumps withcorners will be discussed in 4. In 5, concluding remarks based on the present studywill be outlined.

2. Experiments, ow visualization and observationsThe experimental set-up consists of a closed-loop water-jet system, comprised

of a centrifugal pump (0.5 h.p., head 30/6 m, capacity 15/40 l min 1, 2800 r.p.m.), tworotameters calibrated in the range of 1 .67 10 5 1.67 10 4 m3 s 1 and 1 .67 10 53.33 10 4 m3 s 1, and circular tubes of brass and stainless steel (in the diameter range410 mm). These tubes can be used as nozzles having a length to diameter ratio of 150200, so as to ensure a fully developed ow condition at the exit. The jet issuingfrom the nozzle is made to fall on a square glass plate of dimensions 1 m 1m, anda thickness of 8 mm, mounted on four levelling screws. The edge of the glass plate ischamfered with a radius of approximately 4 mm on the top face, for smooth drainageof the liquid. The set-up is fabricated such that that the jump can be viewed andphotographed from underneath of the glass plate. Calibration of the experimentalset-up has been performed by employing benchmark (experimental) data available inthe literature for circular hydraulic jumps. Figure 2 compares the present experimentalresults with the experimental results of Arakeri & Rao (1996). The gure also showssome of the theoretically predicted estimates of the radius of the jump, using thescaling relations developed by Bohr et al. (1993), Godwin (1993) and Brechet & N eda

(1999), under identical conditions, for the sake of completeness. As can be seen fromgure 2, our experimental results are in good agreement with theoretical predictionsobtained from the scaling relations developed by Bohr et al. (1993), and experimentalresults of Arakeri & Rao (1996).


4/17


5 10 15 20 250

40

80

120

160

200

240

R j

( m m

)

Q 10 5 (m 3 s 1)

Figure 2. The radius of the circular hydraulic jump ( R j ) as a function of the volume ow rateof water ( Q ). , experiments (nozzle diameter, d =10 mm) (Arakeri & Rao 1996); , theory(Bohr et al. 1993); , theory (Brechet & Neda 1999); , theory (Godwin 1993); , presentexperiments ( d = 10mm).

Figure 3. The jump prole is measured using graph paper. A shadow of the jump andimpingement point (indicated with an arrow) is clearly visible.

Flow visualization is carried out using a digital camera (SONY DSC-F717, SonyElectronics NJ). A 1000 W light source is used for illumination from the top. Theimpingement point and the inner edge of the separation eddies are clearly visible,signifying the pertinent jump locations, and are marked on transparent graph sheets(containing square meshes of 1 mm resolution) pasted on the underside of the glassplate (gure 3). From our ow visualization experiments, we summarize that the jumpproles due to obliquely inclined impinging jets can be broadly catagorized as follows:(a) jumps bounded by a smooth curve (jumps formed when > 25 ), and (b) jumpswith corners (jumps formed when 25 ). As becomes less than 90 , the radial

symmetry of the hydraulic jump is lost and it assumes an oblate shape. However,the curve describing this jump is smooth, and we can identify an axis of symmetrypassing through the point of impingement ( = 0) and = , where is the azimuthallocation of a point on the hydraulic jump prole (see gure 1 b). The jump prole


5/17


(b)

(c) (d )

(e)

(a)

Figure 4. Hydraulic jumps at different ow rates, and nozzle inclination angles. ( a) Q = 3 .33 10 5 m3 s 1, d =10 mm, ( b) Q = 1 .17 10 4 m3 s 1, d =10 mm, ( c) Q = 3 .33 10 5 m3 s 1,d = 8 .8mm, n = / 4, (d) Q = 1 10 4 m3 s 1, d = 8 .8mm, n = / 4, (e) unique hydraulic jumpobserved at n = 18 , H = 9 mm, d = 6 .45mm and Q = 4 .17 10 5 m3 s 1.

elongates progressively with an increase of V , and decrease in . It can be noted herethat for > 25 , changes in V can merely alter the slenderness of the jump prole,without creating any corner. However, for 25 , in our experiments, a series of unique jump proles are observed. These proles have one or more corners, and theirshapes are highly sensitive to the variation of V . Typical hydraulic jumps, both dueto normal and oblique impinging jets are depicted in gure 4. The hydraulic jumps atlower ow rates are observed to be steady and distinct (gure 4 a , c). However, jumpsat relatively higher ow rates are unsteady and ill-dened (gure 4 b, d). During ourexperiments, the jet Reynolds number ( Re d )is observed to be in the range of 4000 to25 000. Within this regime, transition to turbulence does not occur. However, beyondRe d = 26 000 it is observed that the circular hydraulic jump loses its radial symmetryand the corresponding non-circular hydraulic jump loses its axial symmetry. Analysisof hydraulic jumps formed as a consequence of oblique impingement of such turbulentliquid jets, however, remains beyond the scope of present study, and therefore, is not

further discussed here.In contrast to the circular hydraulic jumps, it is found that for inclined impinging jets, the drop height, H , implicitly plays a consequential role in determining the sizeand shape of the non-circular hydraulic jump. Gravity effects deect the jet in such


6/17


cases, so that the jet angle, ( ), becomes greater than the nozzle angle ( n ), as canbe seen in gure 2( b). This effect is more pronounced for larger drop heights and atlower ow rates. It can be shown that , n and V are related as:

= tan 1 V

2 sin2 n + 2 gH 1/ 2

V cos n . (2.1)

The value of obtained using (2.1) is used in further calculations reported in thisstudy. In this context, once this effective angle of jet obliquity is ascertained, anyfurther signicance of the drop height, H , becomes practically irrelevant in terms of imposing an explicit inuence on the mechanism of the hydraulic jump formed.

3. Theoretical analysis

3.1. Characterization of the stagnation point for jumps bounded by a smooth curveThe present theoretical analysis borrows its basic outline from the earlierconsiderations of Bradshaw & Love (1959), Scholtz & Trass (1970), Looney & Walsh(1984), Bouainouche, Bourabaa & Desmet (1997), and Phares, Smedley & Flagan(2000). In effect, in the present study, an inviscid outer-ow analysis is coupled withthe boundary-layer approximations for prediction of radial locations of non-circularhydraulic jumps. The analysis primarily deals with the consequence of an obliqueimpingement of a circular jet of radius r0 (diameter, d ) impinging on a stationeryhorizontal plate with V as the velocity of jet impingement. After impingement, theuid ows radially with velocity V r . The distribution of this radial ow is uneven, the

ux being the highest in the forward ow direction and the lowest in the backwardow direction. The stagnation point, here, is considered to be the point of intersectionof the separation streamline with the plane of the impinging surface (gure 5 a). Thesection of this jet, on a plane parallel to the plate, turns out to be an ellipse, of majoraxis (2r0/ sin ), and minor axis (2 r0), as shown in gure 5( b). Figure 5( c) shows thetop view of the radially spreading ow, along with the section of the jet on the targetplate. The origin of the radial ow is located at the stagnation point, S . Assuminga steady ow and neglecting the effects of gravity, surface tension and viscosity, wecan write an expression for mass conservation across the shaded element depicted ingure 5(c) as

(r ed ) r e2 (V sin ) = rp d hV r , (3.1)

where, re = re(, ) is the polar radius of the envelope of the impingement region,and rp is the radial distance of a point located on the plate (relative to the point S ),and h is the lm thickness. Further, from mechanical energy balance, it follows thatthe velocity V is equal to V r . Hence, (3.1) can be expressed as

hr p = r 2e2

sin . (3.2)

Now, equating the forward momentum (in the X -direction) transmitted by the jetwith the net X momentum generated in the radial ow, we obtain

r 20 V 2 cos =

2

0(hr p ) V 2r cos d, (3.3)


7/17


d

r p

r p + d r p

r er e + d r e

S oS s r 0/sin

Impingementregion

(c)

Separation streamline

Stagnation point

V

= 0

V r

=

(a)

X

(b)

2r 0/sin

2r 0

S sS

r e()

S

O

Figure 5. (a) The stagnation point, the point of intersection of the separation streamline withthe plane of the impinging surface. ( b) The section of the circular jet, on a plane parallel tothe plate is an ellipse. ( c) The top view of the radially spreading ow along with the section of the jet on a plate.

where is the density of liquid. From (3.2) and (3.3), we obtain

cot =

0

r er0

2

cos d. (3.4)

Further, the polar equation of the outer boundary of the impingement zone is givenas

r er0

sin 2

+r er0

cos S sr0

2

sin2 = 1 , (3.5)

where S s is the stagnation point shift (i.e. the distance of the stagnation point fromthe geometrical centre of the jump prole, O ). Obtaining an expression for r e /r 0 from(3.5) and using the same in (3.4), it can be shown that

S s = r0 cot . (3.6)

It can now easily be inferred from the above analysis that the polar radius of thenon-circular impingement zone, with respect to the stagnation point (which is anupstream focus of the impingement zone) is given by

r e (, ) = r0 sin

1 + cos cos . (3.7)

3.2. Radial locations, R j ( ), of non-circular hydraulic jumpAs mentioned in 3.1, the impingement zone of the inclined circular jet can bemodelled as an impingement zone of an equivalent vertical elliptic jet, which is


8/17


approximately of the size of the elliptic jet itself. The radially spreading ow in thiscase will be three-dimensional in its nature. This three-dimensional nature of theow can be modelled effectively as a pseudo two-dimensional one by consideringthe coordinate r itself as a function of . Further simplications are made byassuming that surface tension effects are negligible in comparison to gravity effects.Moreover, gradients along the axial directions are assumed to be much more dominantin comparison to those along the radial directions, consistent with the prevailinglength scales. Using these considerations, the pertinent equations of continuity andmomentum in cylindrical coordinates can be written as:

V rr

+ V r

r+

V zz

= 0 , (3.8a)

V rV rr

+ V zV rz

= gdhdr

+ 2V rdz2

, (3.8b)

where V r (r, z ) and V z(r, z ) are the components of the velocity in the radial and verticaldirections, respectively, h (r ) is the thickness of the uid layer, and g is the accelerationdue to gravity. Equations (3.8 a) and (3.8b) are consistent with the following boundaryconditions:

V r (r, 0) = 0 , (3.9a)

V z(r, 0) = 0 , (3.9b)where r = r ( ).

V rz z= h(r )

= 0 , (3.9c)

r h

0V r (r, z ) dz = q , (3.9d)

where q = Q/ 2 , Q being the volume ow rate (as the product of r hV r happens to bean invariant function with respect to ). The coordinate r , which is itself a functionof , is measured with respect to the source of the radially spreading ow.

Equations (3.8 a) and (3.8b) are now re-written in a dimensionless form using typicalscales for the radial velocity V r, vertical velocity V z, radius r and thickness of theliquid layer z, with the following substitutions:

V r = V r, = q 1/ 8 1/ 8g 1/ 8, (3.10a)

V z = V z, = q 1/ 4 3/ 4g 1/ 4, (3.10b)

r = r , = q 5/ 8 3/ 8g 1/ 8, (3.10c)

z = z, = q 1/ 4 1/ 4g 1/ 4. (3.10d)Rescaled equations (dropping asterisks) are then obtained as

V rr

+ V r

r+

V zz

= 0 , (3.11a)

V rV rr + V z

V rz =

dhdr +

2V rdz2 . (3.11b)

The boundary conditions in terms of rescaled quantities can be described as

V r (r, 0) = 0 , (3.12a)


9/17


where r = r ( ),V z(r, 0) = 0 , (3.12b)

V rz z= h (r )

= 0 , (3.12c)

r h(r )

0V r (r, z ) dz = 1 . (3.12d)

Although the boundary-layer analysis break, down in a small region around thepoint of separation, the rest of the ow can be described well by following suchapproximations. The system of averaged equations in the z-direction, however, avoidsshowing the singular behaviour at the point of separation and can adequately describethe ow pattern involving the separation of the boundary layer in an integrated sense.Integrating (3.11 b) over z from 0 to h(r ), and simultaneously exploring the continuityequation (3.11 a), we obtain (after incorporating boundary conditions (3.12 a) to(3.12d)):

1rh

ddr

r h

0V r 2 dz = h

1h

V rz

|z=0 , (3.13)

where the prime denotes differentiation with respect to r .Now, dene:

F 2 = h

0

V rV av

2

dz, (3.14)

where V av is the average velocity, dened as

V av = 1h h

0V r (r, z ) dz. (3.15)

Further, conservation of mass yields

V av hr = 1 . (3.16)

From equations (3.13) to (3.16), we obtain:

V av (F 2V av ) = h 1h

V rz

|z=0 . (3.17)

For further simplications, a judicious choice of the velocity prole must be made.In this context, it can be noted here that the classical von K arm an and Pohlhausenmethod for boundary-layer analysis assumes a velocity prole that is ascertainedpurely by the external ow conditions (Schlichting 1960). Here, however, the thicknessof the uid lm, h(r ), plays a role that is very much analogous to the boundary-layerthickness in a classical viscous ow analysis, in terms of dictating the exact self-similarnature of the velocity prole. Accordingly, one may write

V r (r, z )V av

= f (), (3.18)

where = z/ h (r ) and V av is the average velocity, satisfying (3.16). Since the velocityprole must also satisfy the boundary conditions (3.12 a) to (3.12d) and (3.16), f ()must obey the following restraints:

f (0) = 0 , (3.19)


10/17


100

10 1

10 2

100 101 102 10 110 2 101100 102

101

102

103

V a v

r () r ()

V a vh

(a)

100

10 1

10 2

101

102

103(b)

Figure 6. (a) Numerical solution of the equation (3.25), ( b) typical solution for V av and h,which is indicative of a characteristic discontinuity at r ( ) 1.

f (1) = 0 , (3.20)

1

0f () d = 1 . (3.21)

For example, a quadratic prole

f () = 3 32 2, (3.22)

may be one of the simplest possible polynomial forms for the function f , satisfying(3.19) to (3.21). Then, from (3.14), F 2 = 6 / 5, is obtained as a constant, and (3.17)

becomes65 V av V av = h

3V avh 2

. (3.23)

Any other valid ansatz for f will still lead to the same form as (3.23), except for thenumerical coefficients. There is no qualitative change, however, since all equationsof the type (3.23), corresponding to different prole shapes, f (), can be furthertransformed to a general form:

V av V av + h = V av

h 2 , (3.24)

after suitable rescaling of h , v and r . Combining equations (3.16) and (3.24), we obtain

V av V av 1V av 2r

= 1V av r 2

V av 3r 2. (3.25)

Typical solutions of (3.25) are shown in gure 6. At large enough values of r ( ),the velocity suddenly increases, which corresponds to a sudden drop in height. Thissingularity is interpreted physically as the end of the plate where the water runsoff (Bohr et al. 1993). The above solution for the ow eld after the jump can beconnected to the solution before the jump by means of a Rayleigh shock, whichconserves mass and momentum ux given as follows:

V av 1h 1 = V av 2h 2, (3.26a)

h 1 V av 12 + gh 1

2 = h 2 V av 22 +

gh 22

, (3.26b)


11/17


Velocity prole Approximate value of C83

43

3 0.76209

59

4 0.83

157 37 5 0.85Table 1. Numerical value of constant C (equation (3.27)) for higher-order velocity proles.

0

50

100

180210

240

270

300

330

0

50

100

(a)

0

50

100

150

180210

240

270

300

330

50

100

150

(b)

0

50

100

150

180210

240

270

300

330

0

50

100

150

(c)

theoretical theoretical (CHJ)experimental experimental (CHJ)

Figure 7. Radial locations, R j ( ) of the hydraulic jumps for different values of nozzle angles(n ) and ow rates ( Q ): (a) r0 = 3 .5mm, Q = 5 .83 10 5 m3 s 1, n = 60 , H = 56mm(b) r0 = 4 .35mm, Q = 8 .33 10 5 m3 s 1, n = 45 , H = 80mm, ( c) r0 = 3 .25mm,Q = 5 10 5 m3 s 1, n = 35 , H = 40 mm.

where, subscripts 1 and 2 correspond to values before and after the jump. Sincethe above shock (which can be interpreted as a hydraulic jump, in the present case)occurs very close to r ( ) = 1 (in the dimensionless coordinates), the radial locationof the jump, R j ( ), scales approximately as r, and is given from (3.10 c) as

R j = = C q 5/ 8 3/ 8g 1/ 8. (3.27)

Here, the constant C depends on the velocity prole chosen in (3.14). For a parabolicprole, for example, C turns out to be approximately 0.73. For higher-order proles,satisfying the requisite boundary conditions (equations (3.19) to (3.21)), the respectivevalues of C are shown in table 1. Further, for the present case, q can be expressed as

q = Q(2 )

= r h ( )V r . (3.28)

From equations (3.2), (3.7) and (3.28), we obtain, at rp = r e

q = r h ( )V = r02

2 sin3

(1 + cos cos )2V . (3.29)


12/17


10 20 30 40 50 60 70 80 901.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

Jumpswith

corners

a b

theoretical (3.34) experimental

(deg.)

Figure 8. Aspect ratio (ratio of major axis, a , to minor axis, b) of non-circular hydraulic jumps is independent of V for a xed .

With this value of q, radial locations of the jump R j (, ) are obtained as

R j (, ) = Cr02

2sin3

(1 + cos cos )2V

5/ 8 3/ 8g 1/ 8. (3.30)

As a special case, the radius of the circular hydraulic jump can be obtained by putting = / 2 in equation (3.30), as

R j = Cr 02V

2

5/ 8

3/ 8g 1/ 8, (3.31)

which can also be written as

R j = C q 5/ 8 3/ 8g 1/ 8. (3.32)

Equation (3.32) is essentially the same as that developed by Bohr et al. (1993), whichshows that the instance of a circular hydraulic jump can be obtained as a special casefrom the present generalized theory.

A comparison of experimental results and the present theoretical predictions isshown in gure 7. Experimental results are in extremely good agreement with thecorresponding theoretical predictions for the hydraulic jumps bounded by a smoothcurve. Predictions regarding circular hydraulic jumps (CHJ), ( = / 2), are alsoplotted on the same gure, for effective comparisons. It can be shown that the aspect

ratio (ratio of the major axis, a , to the minor axis, b) of the non-circular hydraulic jump is given byab

= 1

(1 cos2 )5/ 8. (3.33)


13/17


Point O

(b)

(a)

= 0 =

Figure 9. (a) Illustrative sketch of jumpjet interaction. ( b) Jetjump interaction (encircled)observed during our experiments. This interaction perturbs the ow leading to corneredhydraulic jumps.

Hence, the aspect ratio is a function of only, and is independent of V , as mentionedin 2. Variations in aspect ratio with , as obtained using (3.33) and from ourlaboratory experiments, are shown in gure 8.

4. Hydraulic jumps with cornersA series of jump proles with altogether different shapes emerge for 25 . At

these values of , the elliptic impingement zone elongates substantially such thatthe jump prole at the extreme upstream position ( = 0) makes a contact withthe impinging jet (see gure 9). This conuence, termed the jumpjet interaction, isdependent on parameters , V and d . The upstream location of the impingementzone, which now coincides with the extreme upstream of the jump prole, becomessomewhat pointed, as depicted in gure 9 (refer to the point O in gure 9a). Fromthis point, two identical branches of jump emerge. At the vicinity of the impingementzone, the uid undergoes a shooting ow, with a relatively high velocity. This makestwo branches of the jump diverge from each other. However, the two branches arecurved, as is the boundary of the impingement zone itself. Further downstream, the

two branches come closer. Depending on the distance between the branches and theirstrengths, different interactions may be possible. These interactions can be termedas the jumpjump interactions. At a relatively low impingement velocity, the jumpprole shape more or less conforms to isosceles triangular shapes (gure 10( a (i), (ii)


14/17


(a )(i)

(b)(i)

(c)(i)

(c)(ii)

(a )(ii)

(b)(ii)

Figure 10. Hydraulic jump with corners: ( a )(i), (b)(i) and ( c)(i) are type I, type II and type III jump proles, respectively, ( a )(ii) (n = 17 .5 , Q = 5 10 5 m3 s 1, d = 6 .48mm, H = 11 mm),(b)(ii) (n = 17 .5 , Q = 8 .33 10 5 m3 s 1, d = 6 .48mm, H = 11 mm), and ( c)(ii) (n = 17 .5 ,Q = 1 .25 10 4 m3 s 1, d = 6 .48mm, H = 11 mm) are the corresponding cornered jumpsvisualized in our experiments.

(type I jumps)), the vertex of the triangle being located at the extreme upstream pointof the jump prole. Additionally, we observed the formation of wave structures anddisturbances beyond the base (side opposite to the extreme upstream) of the triangle,as can be seen in gure 10( a )(i).

With an increase in V , the extreme upstream of the jump becomes more pointed.The two equal sides of the jump prole elongate and become curved. The base takesa bow shape and its length decreases. The typical form observed can be described asa tear drop shape (gure 10 b(i), (ii)) (type II jumps). With further increases in V , thetwo branches of the jump prole, emerging from the extreme upstream point comecloser to each other. At one point, the third branch (i.e. the branch perpendicularto the line of symmetry) disappears and the other two branches intersect eachother, giving the jump prole a typical shape. The shape now resembles that of a sh (gure 10 c(i), (ii)) (type III jump). With any increase in V beyond this, theprole elongates and encompasses a greater area. However, the jump prole shaperemains unaltered. Figure 11 depicts the qualitative changes in the hydraulic jump

proles with the corrosponding variations in and V . Disturbances and waves of asecondary nature have also been observed beyond the third branch and at the point of intersection of the branches (except at the extreme upstream point)(see gures 10 a (ii)and b(ii)).


15/17


15 20 25 75 80 85 90

2

3

4

C i r c u

l a r

h y d r a u

l i c j u m p

T r a n s

i t i o n r e g i o n

Jumps bounded bya smooth curve

V ( m

s 1 )

Type I

Type II

Type III

(deg.)

Figure 11. Phase diagram for hydraulic jumps due to obliquely impinging circular liquid jeton a at horizontal surface, ( d = 6 .45 mm, H = 9 mm).

A

BS inc

S ref

S ref

O

A

B

S

(a )

S inc

O S m

(b)(c)

1 2

1 2

Figure 12. Shock reections: ( a) regular reection, and ( b) Mach reection, S inc and S ref areincident and reected shocks, respectively, and S m is the Mach stem. ( c) Waves and structuresof a secondary nature in shock reections.

Qualitative phenomenological explanations for the formation of such non-intuitivehydraulic jump proles can be provided by appealing to the similarity between ahydraulic jump and a compression shock (Preiswerk 1940). When two oblique wavesmeet each other, the axis of symmetry passing through the meeting point conceptuallyacts as an impervious wall. One of the shocks may be considered to be incident onthat wall, while the other is reected. Two types of reection, namely regular reectionand Mach reection (gure 12), have been reported to characterize these situations. InMach reection, the incident wave and the reected wave meet at a point away from

the axis of the symmetry. A third wave, known as a Mach stem, joins this point withthe axis of symmetry. The Mach stem is, in general, curved at the triple point. Theentire stem may also be curved, depending on the characteristics of the interactingwaves. Further, waves and structures of a secondary nature are observed downstream


16/17


of the Mach stem (gure 12 c). Similar interactions between discontinuity waves inshallow-water ows have been observed by Gilmore, Plesset & Crossley (1950).

The hydraulic jump proles with corners (gure 10) can be explained in light of regular and Mach reections mentioned as above (refer to gure 12). Formation of thetriangular jump prole (type I) corresponds to the case of a Mach reection, wherethe base of the triangle is nothing but a Mach stem. With further development,the Mach stem becomes shorter, as observed in type II proles. Eventually, acase of regular reection occurs as the ow rate is further increased, leading toa complete disappearance of the Mach Stem in type III proles. Waves and structuresof a secondary nature (see gures 10 a (ii) and b(ii), similar to those encounteredin compressible gasdynamics beyond the shock wave interactions, have also beenobserved during the present experiments.

5. Summary and conclusions

In this paper, we have studied hydraulic jumps that occur when circular impinging jets fall on a smooth horizontal plate at different angles of obliquity. Changes in theow eld of an impinging jet owing to its inclination are analysed from both theoreticaland experimental perspectives. It has been revealed that the circular impingementzone deforms to an equivalent elliptic one. The stagnation point, consequently, shiftsto the upstream focus of the elliptical impingement zone. The analysis for the radiallocations of the non-circular hydraulic jump prole (jumps bounded by a smoothcurve), based on this conjucture, gives satisfactory predictions, as evident from thefavourable comparisons with experimental observations. For jumps with corners,which form at lower angles of jet inclination, strikingly contrasting proles can beobtained, for which a phenomenological explanation is provided by drawing analogiesfrom shock-wave interactions.

REFERENCES

Abramovich, G. N. 1963 The Theory of Turbulent Jets (ed. L. H. Schindel). MIT Press.Arakeri, J. H. & Rao, A. 1996 On radial ow on a horizontal surface and the circular hydraulic

jump. J. Indian Inst. Sci. 76 , 7391.Beltos, S. 1975 Oblique impingement of circular turbulent jets. J Hydraul. Res. 14 , 1736.Blackford, B. L. 1996 The hydraulic jump in radially spreading ow: a new model and new

experimental data. Am. J. Phys. 64 , 164169.Bohr, T., Dimon, P. & Putkaradze, V. 1993 Shallow-water approach to the circular hydraulic

jump. J. Fluid Mech. 254 , 635648.Bouainouche, M., Bourabaa, N. & Desmet, B. 1997 Numerical study of the wall shear produced

by the impingement of a plane turbulent jet on a plate. Intl J. Numer. Meth. Heat Fluid Flow.7 , 548564.

Bradshaw, P. & Love, E. M. 1959 The normal impinging jet of a circular air jet over a at surface. ARC R &M 3205. Aero. Res. Council, UK.

Brechet, Y. & N eda, Z. 1999 On the circular hydraulic jump. Am. J. Phys. 67 , 723731.Bush, J. W. M. & Aristoff, J. M. 2003 The inuence of surface tension on the circular hydraulic

jump. J. Fluid. Mech. 489 , 229238.Craik, A., Lathman, R., Fawkes, M. & Gibbon, P. 1981 The circular hydraulic jump. J. Fluid.

Mech. 112 , 347362.Gilmore, F. R., Plesset, M. S. & Crossley Jr, H. E. 1950 The analogy between hydraulic jumps

in liquids and shock wave in gases. J. Appl. Phys. 21 , 243249.Glauart, M. B. 1956 The wall jet. J. Fluid. Mech. 1 , 625643.Godwin, R. 1993 The hydraulic jump (shocks and viscous ow in the kitchen sink). Am. J. Phys.

61 , 829832.


17/17


Hansen, S. H., Horl uck, S., Zauner, D., Dimon, P., Ellegaard, P. & Watanabe, S. 1997Geometrical orbits of surface waves from a circular hydraulic jump. Phy. Rev. E 55 , 70487061.

Higuera, F. J. 1994 Hydraulic jump in a viscous laminar ow. J. Fluid Mech. 274 , 6992.Higuera, F. J. 1997 The circular hydraulic jump. Phys. Fluids 9 , 14761478.Ishigai, S., Nakanishi, S., Mizunao, M. & Imamura, T. 1977 Heat transfer of the impinging round

water jet in the interference zone of lm owing along the wall. Bull. JSME 20 , 8592.Liu, X. & Lienhard, J. 1993 The hydraulic jump in circular jet impingement and in other thin

liquid lms. Exps. Fluids 15 , 108116.Looney, M. K. & Walsh, J. J. 1984 Mean-ow and turbulent characteristics of free and impinging

jet ow. J. Fluid Mech. 147 , 397429.Nakoryakov,V., Pokusaev, B. & Troyan, E. 1978 Impingement of an axisymmetric liquid jet on a

barrier. Intl J. Heat Mass Transfer 21 , 11751184.Olsson, R. G. & Turkdogan, E. T. 1966 Radial spread of a liquid stream on a horizontal plate.

Nature 211 , 813816.Phares, D. J., Smedley, G. T. & Flagan, R. C. 2000 The wall shear stress produced by the normal

impingement of a jet on a at surface. J. Fluid Mech. 418 , 351375.

Preiswerk, E. 1940 Applications of the methods of gas dynamics to water ows with free surfaces. NACA TM 934 and 935.Rajaratnam, N. 1976 Turbulent Jets. Elsevier.Rao, A. & Arakeri, J. H. 1998 Integral analysis applied to radial lm ows. Intl J. Heat Mass

Transfer 41 , 27572767.Rubel, A. 1981 Computations of the oblique impingement of round jets upon a plane wall. AIAA

J. 19 , 863871.Rubel, A. 1982 Oblique impingement of a round jet on plane surface. AIAA J. 20 , 17561758.Schlichting, H. 1960 Boundary Layer Theory. McGraw-Hill.Scholtz, M. T. & Trass, O. 1970 Mass transfer in nonuniform impinging jet. AIChE J. 16 , 8290.Sparrow, E. M. & Lovell, B. J. 1980 Heat transfer characteristics of an obliquely impinging

circular jet. Trans. ASME C: J. Heat Transfer 102 , 202209.Stevens, J. & Webb, B. W. 1991 The effect of inclination on local heat transfer under an axisymmetric

free liquid jet. Intl J. Heat Mass Transfer 34 , 12271236.Tong, A. Y. 2003 On the oblique impingement heat transfer of an oblique free surface plane jet.

J. Heat Mass Transfer 46 , 20772085.Taylor, G. 1960 Formation of thin at sheets of water. Proc. R. Soc. Lond. A 259 , 117.Watson, E. J. 1964 The spread of a liquid jet over a horizontal plane. J. Fluid Mech. 20 , 481499.Yokoi, K. & Xiao, F. 2002 Mechanism of structure formation in circular hydraulic jumps: numerical

studies of strongly deformed free-surface shallow ows. Physica D 161 , 202219.

hydraulic jumps due to oblique impingement of circular liquid jets on a flat horizontal surface

Documents