Wang Yongtao, Cui Li and Wang Qingguo
Dalian University of Technology, Danlian 116024
Abstract: The article applies Newton's general similarity criterion to the distorted model. The results of gravity force similarity, drag force similarity and pressure force similarity is identical with the conclusions which is derived from relevant differential equations of liquid flow, and we inquire into selected limits of the distorted ratio, the simulation of roughness coefficient of distorted model by means of hydraulic test.
Keywords: distorted model, similarity criterion, distorted ratio, simulation of roughness coefficient
In water conservancy, energy and triffic project, in order to verify the reasonableness of project design, or to study possible problem in project design and running, we often use model test. In gemeral, the similarity of geometrically normal model is very good. Because of complexity of test condition and practical problem, it is impossible that all models can realize normal, at present, similarity theory and applied research of distorted model aren't very perfect, similarity criterion is all derived from relevant differential equations of liquid flow, so, the physical concept isn't very clear, and we'll get into a great trouble to analyse some concret matter. In this paper, according to the basic fact of geometrical distortion, by means of motion term and dynamics term analysed, the concept of distortion of force is introduced, and the general similarity criterion in practice is derived from simple Newton's general similarity criterion, the results is identical with the conclusion which is dirived from relevant differential equations of liquid flow.
The basic fact of distorted model is that the geometrical scale isn't equal in a horizontal and vertical direction, what influence will this have on motion term and dynamics term?. It is very important for the derivation of similarity criterion.
Suppose the scale of horizonal length is ll, the scale of vertical length is lh , the distorted ratio is e : e = ll /lh, In addition , the scale of gravity acceleration g is lg=l. The dimension and scale of motion term are derived as follws: suppose u is horizonal velocity ,v is vertical velocity, then,
lu = ll /lh ; lV = lh /lt and lu/lV =(ll /lh)/(lh /lt )= ll /lt=e
Suppose al is horizonai acceleration; ah is vertical acceleration , then :
lal = ll /lt2 ; lah = lh /lt2 and lal /lah=ll /lh=e
The ratio of horizonal inertia force to that of
vertical:
Where : r ¾ liquid density ; /, h ¾ geometrical length ; u ¾ liquid horizonal velocity.
lFl /lFh=ll /lh=e (2-1)
Deserving to be noted , because of lg=l , gravity force will not synchronize with the force derived from liquid motion, this is a chief drawback of distorted model, we can see its effect below the derivation of similarity law.
It is known by the Newton's similarity
criterion, the ratio of acting force to inertia force is equal in the model and prototype,
i.e., Nem =Nep
Gravity force is divided by the vertical inertia force, we get:
(3-1)
When gravity force divides by the horizonal inertia force, since the horiaonal inertia force isn't identical with that of the vertical in the model, there is a distorted ratio lFl /lFh =e :, thus in Newton number of model, we must apply FHm =FLm/e_, simplify, we obtain :
lu2 = ll /e=lh (3-2)
Eq.(3-l),(3-2) is identical with the similarity law derived from the vertical and horizonal differential equations of liquid flow respectively.
If it is simultaneously satisfied that relative action of liquid gravity is similar in a vertical and horizonal direction, i.e., Eq. (3-1),(3-2) is satisfied simultaneously, then : lu2 = lh =ell, i.e. ll /lh=e , It is in contradiction with the fact of lh /ll=e. therefore, it is impossible that distorted model satisfied gravity force similarity, the reason is that gravity force isn't identical with the force derived form liquid flow during the simulation, so the relevant similarity criterion of gravity force isn't satisfied perfectly.
However, when the water flows gently, the vertical motion isn't clear, then Eq. (3-2) can be taken as the gravity similarity criterion of model design to simulate approximately the flow. If one part of model flow is principal in a vertical direction, then the part is designed by Eq. (3-1), When there are the horizonal and vertical flow simultaneously, we can only design by normal model. 3-2 Drag force similarity criterion (1) Flow in laminar flow layer
Because of the particularity of river section, the drag can be divided into two parts of river bed and bank, Tb, and Ts,
Suppose the water flows gently in the channel, we can approximately substitute the horizonal velocity u for the fluid velocity.
From: [ Nes]p= [Nes]m , we get the drag force similarity criterion of bank,
(3-3)
Where, v is kinematic viscosity, v = m./r , similarly, we get the drag similarity criterion of river bed:
(3-4)
Eq. (3-4) is identical with the results dirived from differential equations of liquid flow. If the drag similarity of bank and river bed is satisfied simultaneously, then it is asked for that
Eq. (3-3),(3-4) is satisfied simulaneously, i.e., ll/lh=lv/lu
So,e2=l, e=1 ,it is in contradiction with e¹1. Hence, it is impossible that the lamina flow drag similarity of river bed and bank is satisfied simultaneously, usually, it is designed by Eq. (3-4),so the error of ratio of bank drag to inertial force is:
i.e. the ratio of bank drag to inertial force is s2 times as much as the prototype, and the boundary drag of model is enlarged very much.
The drag is divided by the inertia of same direction, from the Newton's general similarity criterion, we get:
(1) Similarity criterion of bank drag
(3-5)
Where: ns
¾roughness coefficient of bank,
;
Rs¾hydraulic radius of bank; R^. = 5/2
(2) Drag similarity criterion of river bed
Where: nb¾roughness coefficient of river bed ; Rb¾hydraulic radius of river bed ; Rb=h
lnb= ll –1/2lh-2/3 (3-6)
Eq. (3-6) is identical with the drag similarity criterion dirived from one dimensional open channel nonuiform flow differential equations, but Eq. (3-6) is very clear, so, if the drag similarity of bank and river bed is satisfied simultaneously, it must satisfy
(3-7)
Eq.(3-7) can control the value of distorted ratio e in practical project, however, a lot of research work is experiencial now, and much research is needed, for the wide and shallow channel, the design is basically satisfied by Eq. (3-6).
The general expression of hydrodynamic pressure force is : P = p×A ,
Where, p ¾ hydrodynamic pressure ; A¾ area of acting force. Hydrodynamic pressure force is divided by the inertia force of same direction, from the
Newton's general similarity criterion, we obtain: the horizonal hydrodynamic pressure force
similarity criterion
(3-8)
The vertical hydrodynamic pressure force similarity criterion
(3-9)
Eq.(3-8),Eq.(3-9) is identical with the results derived from the horizonal and vertical differential equations of liquid flow, but if both Eq. are satisfied simultaneously, it is needed that e=1 , this is in contradiction with e ¹1 , in order to grasp principal contradiction, we must do flexibly in practice.
In front of the paper, we have derived that similarity criterion number of several main force is equal, it is impossible to be carried out completely in distorted model now, we must grasp principle contradiction and main simulant direction in practice, the selected limits of distorted ratio, and treatment of roughness coefficient m me simulation of turbulent drag. We say a few words about domestic experience and our practice here.
In general, a distorted model only asks to solve the problem that mean hydraulic factor satisfies engineering needs, i.e.,the mean hydraulic factor is similar, but lc ¹ 1, ll = 1.therefore, to a certain extent, the velocity distribution will be distorted, the more distorted ratio is big, the more it is distorted, usually, it is right that e is smaller than 5.0,
The rough value of distorted ratio is given by Claus's formula[3] e£ B/10h, where , h is water depth, B is the width of velocity field.
In the model test
of river works, the roughness coefficient simulation can't be neglected,
specialy, after tailwater course cleared, the major energy loss transforms local
head loss (in front of excavation ) into frictional head loss. The scale of
roughness coefficient n of river bed:
ln=ll-1/2lh 2/3, the scale of Chezy's coefficient c : lc=(lllh )1/2
During processing a model, we can select the sands and stones of various kinds of size to lay or various kinds of rough materials to simulate roughness coefficient by experiment. But the problem is that sometimes river depth is small, increasing roughness on bed affects states of flow, and it isn't easy to be carried out; sometimes we get the Chezy's coefficient cm’ of model material by hydraulic test, so ,on the basis of prototype datum, we gain the model scale lc’ (it is different with discharge). At large, It is lc/lc’ times as small as the scale lc of satisfying frictional drag similarity, therefore, at the moment, the discharge scale lQ’ is correspondingly lc/lc’ times as small, then,
lQ’=lQ/(lc/lc’)=ll0.5lh2.0 lc’We do experiment of different slope with the sands of both size, the result shows to gain the same water depth, small roughness coefficient can be carried out by means of the discharge multipplied by lc/lc’, shown in Fig. 1, Fig. 2. In the limits of test discharge, if in the drag square area, the discharge changes within Qmax/Qmin £5, the change of lc’ isn't big, the drawback of roughness coefficient similarity simulation can be made up by enlarging model discharge[2][4][5][6], i.e., by means of model practical surveying roughness coefficient, we gain adjusted scale of discharge under different discharge, {ln}®{lc’}®{lQ’}, after compared with the prototype surveying datum, both are very identical[6].
(1) This paper applies Newton's general similarity criterion to the distorted model, the results of gravity force similarity, drag force similarity and pressure force similarity is identical with the conclusions dirived from relevant differential equations of liquid flow, the physical concept is clear, and the analytical method is brief, however, it should be noted that, during apping Newton's similarity criterion, the force in Newton number is in the same direction;
(2) Gravity force similarity and turbulent drag similarity are satisfied in the design of distorted model, it is suitable for the tranquil flow, wide and shallow river in drag square area;
(3) For the limits of distorted ratio, on the basis of author's information, it is shown that e =f(B, h), in general, within e is smaller than 5.0, the simulation is good, the velocity distribution isn't distorted very much;
(4) The simulation of model roughness coefficient is very complicated, one is that river bed is complicated, two is that roughness coefficient itself is changeable, it is changeable with discharge, for small roughness coefficient, we can make flow profile similarity by enlarging model discharge.
Fig.1
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