(1) What relationship should coefficients A、B、C、D satisfy? (2) The stream function of the flow field. 已知流场的速度分布为

vx=Ax+By vy=Cx+Dy 若流体不可压缩，且流动无旋，试问

(1) 系数A、B、C、D应满足怎样的关系？ (2) 求流场的流函数。

( (1) A=?D，B=C；(2) ?=B(y2?x2)/2+Axy )

4.16 There is a fixed point vortex of circulation ? and distance a to a stationary wall. Find the velocity potential function of the flow and pressure distribution on the wall.

有一环量为?的固定点涡，离一静止壁面的距离为a。试求流动的速度势和 壁面上的压强分布。 (

，

)

4.17 Given the velocity of an incompressible planar potential flow as vx=3ax2?3ay2, vx=vy=0 at point (0，0), find the volume flowrate passing the connecting line of points (0，0) and (0，1).

已知不可压缩平面势流的速度分布为 vx=3ax2-3ay2，在(0，0)点上vx=vy=0，试求通过(0，0)、 (0，1)两点连线的体积流量。( qv=a )

4.18 A two-dimensional flow field is formed by adding a source at the origin of the coordinate system to the velocity potential ?=r2cos2?

Locate any stagnation points in the upper half of the coordinate plane. (0????) 一二维流场是在速度势?=r2cos2?上于坐标原点处叠加一点源而形成。试在坐

0.5

标平面上半部确定任一驻点的位置。( ?s=?/2，rs=(m/4?))

4.19 The stream function for a two-dimensional, incompressible flow field is given by the equation

?=2x-2y

where the stream function has the units of m2/s with x and y in meter.

(1) Sketch the stream function for this flow field, indicate the direction of flow along the streamlines;

(2) Is this an irrotational flow field?

(3) Determine the acceleration of a fluid particle at the point x=1m and y=2m. 二维不可压流场的流函数为

?=2x-2y 式中x、y的单位为m，流函数单位为m2/s。

(1) 做出流线分布图，标明流线方向；

(2) 流动是否无旋？

(3) 确定点(1,2)处流体质点的加速度。 ( (1) ?=2x?2y，(2) 无旋，(3) a=0)

4.20 The stream function for the flow of a nonviscous, incompressible fluid in the vicinity of a corner (see Fig. 4-27) is

?=2r4/3sin(4?/3). Determine an expression for the pressure gradient along the boundary ?=3?/4.

一无粘不可压流体在如图4-27所示的转角附近流动的流函数为?=2r4/3sin(4?/3)，

试确定沿边界?=3?/4的压强梯度表达式。 Fig. 4-27 Problem 4.20

(

)

4.21 Two sources are located at points (1，0) and (-1，0)，their source strength are all 4?，find the velocity at points (0，0)、 (0，1)、 (0，-1)、(1，1).

位于(1，0)和(-1，0)之两个点源，其源强度均为4?，试求在(0，0)、 (0，1)、

(0，-1)、(1，1)处的速度。

( vx=vy=0；vx=0，vy=2；vx=0，vy=?2；vx=4/5，vy=12/5)

4.22 A source with strength 20m2/s is located at point (-1，0)，another source with strength 40m2/s is located at point (2，0). Given the pressure at the coordinate base point of the overlapped flow field is 100Pa，the fluid density is 1.8kg/m3，find the velocity and pressure at point (0，1) and (1，1).

强度为20m2/s的点源位于(-1，0)，强度为40m2/s的点源位于(2，0)，已知叠加流场在坐标原点处的压强为100Pa，流体的密度为1.8kg/m3，求在点(0，1)和点(1，1)处的速度与压强。

(

?p)

Problems

5.1 A 1:25 scale model of an airship is tested in water at 200C. If the airship travels 5m/s in air at atmospheric pressure and 200C, find the velocity for the model to achieve dynamic similitude. Also, find the ratio of the drag force on the prototype to that on the model. The densities of water and air at these conditions are 1000kg/m3 and 1.2kg/m3 respectively. The corresponding dynamic viscosities of water and air are 10-3N?s/m2 and 1.81?10-5 N?s/m2。

一比尺为1:25的飞艇模型在200C的水中实验。如果飞艇是在200C、大气压力下的空气中以5m/s的速度飞行，为了达到动力相似，求模型的速度。并求原型与模型的阻力之比。已知在实验条件下水与空气的密度分别为1000kg/m3 及 1.2kg/m3，相应的动力粘度为10-3N?s/m2 与 1.81?10-5 N?s/m2。(8.29m/s，0.273)

5.2 A scale model of a pumping system is to be tested to determined the head losses in the actual system. Air with a specific weight of 0.085kg/m3 and a viscosity of 3.74?10-7 m2/s is used in the model. Another fluid with a specific weight of 62.4kg/m3 and a viscosity of 2.36?10-5m2/s is used in the prototype. The velocity in the prototype is 2m/s. A practical upper limit for the air velocity in the model to avoid compressibility effects is 100m/s. Find the scale ratio for the model and the ratio of the pressure losses in the prototype to those in the model.

对一抽送系统的缩小模型进行实验以确定原型的水头损失。原型所使用的空气重度为0.085kg/m3 、粘度为3.74?10-7 m2/s。另一种重度为62.4kg/m3、粘度为2.36?10-5m2/s的流体用于原型实验。原型速度为2m/s。为了避免压缩性效应，模型中空气流速的实际上限为100m/s。求模型的比尺与原型及模型中压强损失比尺。(0.294)

5.3 To study the flow of a spillway with a model of the length scale ratio kl=1:20. It is known that Fr of the prototype and model are equal, flowrate of the model is measured as 0.19m3/s. Find the flowrate of the prototype.

用模型研究溢流道的流动，采用的长度比例系数kl=1:20，已知原型与模型的

3

Fr相等，测得模型上的流量为0.19m/s。求原型上的流量。(339m3/s)

5.4 The kinematic viscosity of a fluid in the prototype is ?=15?10-5m2/s, the length scale ratio of the model is 1:5, if let Fr and Eu be the decisive similitude numbers, what is the kinematic viscosity of the fluid in the model ?

原型中流体的运动粘度?=15?10-5m2/s，模型的长度比例系数为1:5，如以Fr和Eu作为决定性的相似准数，模型流体的运动粘度?m应为多少？(1.34?10-5m2/s)

5.5 The sloshing of oil in a tank is affected by both viscous and gravitational effects. A 1:4 scale model of oil with a kinematic viscosity of 1.1?10-4 m2/s is to be used to study the sloshing. Find the kinematic viscosity of the liquid to be used in the model.

油在容器中的晃荡受到粘性与重力的影响。用一比尺为1:4的模型来研究运动粘度为1.1?10-4 m2/s的油的晃荡。求模型中液体的运动粘度。(1.37?10-5m2/s)

5.6 A wind-tunnel test is performed on a 1: 20 scale model of a supersonic aircraft. The prototype aircraft flies at 480m/s in conditions where the speed of sound is 300m/s and the air density is 1.0kg/m3. The model aircraft is tested in a wind-tunnel in which the speed of sound is 279m/s and the air density is 0.43kg/m3. The drag force on the model is 100N. What speed must the flow in the wind-tunnel be for dynamic similitude, and what is the drag force on the prototype?

对一比尺为1:20的超音速飞机模型进行风洞实验。原型飞机在音速为300m/s、密度为1.0kg/m3的空气中以480m/s的速度飞行。模型飞机在音速为279m/s、密度为0.43kg/m3的空气中进行风洞实验。测得模型的阻力位100N。为达到动力相似，风洞的流速应为多少？原型上的阻力是多少？(446m/s，108kN)

5.7 A ventilation pipe of diameter 1m and average flowing velocity 10m/s. Model test is performed on a water pipe of diameter 0.1m, what is the velocity in the water pipe to achieve dynamic similitude? Suppose the pressure and temperature of air and water are all 101kPa and 200C.

直径为1m的空气管道，平均流速为10m/s，现用直径为0.1m的水管进行模

200C。 型实验，为了动力相似，水管中的流速应为多大？设空气和水均为101kPa、

(6.73 m/s)

5.8 In order to predict the drag on a smooth, streamlined object flying in air, a model is designed to test in water. It is known that the length of the prototype is 3m, flies in air at a speed of 10 m/s. The designing length of the model is 50cm, what is the velocity of water? If the drag on model is measured as 15N, what is the drag on the prototype? Suppose the pressure and temperature of the prototype and model are all 101kPa and 200C.

为了预测一光滑流线型物体在空气中的飞行阻力，设计一模型在水中实验。已知原型的长度为3m，以10 m/s的速度在空气中飞行。模型的长度设计为50cm，水流的速度应为多少？若测得模型受到的阻力为15N，原型受到的阻力将是多少？设原型、模型均处于101kPa、200C。(4 m/s，4.05N)

5.9 The height of an automobile is 1.5m, travels in air 200C at a speed of 108km/h. Air in model test is 00C and its flowing velocity is 60m/s. Find the height of the model. If the front resistance in model test is measured as 1300N, what is the front resistance on the prototype automobile when running ?

汽车高度为1.5m，速度为108km/h， 行驶在200C的空气中，模型实验的空气为00C，气流速度为60m/s。求模型试验汽车的高度。如果在模型实验中测得正面阻力位1300N，求实物汽车行驶时的正面阻力是多少。(0.654m，1586N)

5.10 The surface tension of pure water is 0.073N/m, and the surface tension of soapy water is 0.025N/m. If a pure water droplet breaks up in an airstream that is moving at 10m/s, at what speed would the same size soapy water droplet break up? 纯净水的表面张力为0.073N/m，肥皂水的表面张力为0.025N/m。如果纯净水滴在10m/s的气流中破裂，问尺寸相同的肥皂水滴的破裂气流速度为多少？ (5.85 m/s)