曹 伟
论文题目：二维超音速混合层增强混合的研究
作者简介：曹 伟，男，1962年12月出生，1998年03月师从于天津大学周恒教授，于2001年03月获博士学位。
摘 要
未来航空航天技术的发展要求解决超音速流的混合问题。对混合层来说，由线性稳定性理论可知，随着可压缩性影响的的增大，其不稳定性将减弱，不易产生大尺度涡，从而降低了流体的混合能力。仅仅依靠控制其不稳定波的发展，混合的效果很难提高。因此，需要寻找其它激发大尺度结构的控制方法。
1993 Wang & Fiedler 对圆管中的不可压混合层用一种不同于传统的方法对其进行控制，混合效果得以大大增强。本文要研究的一个主要问题是，同样的方法对可压缩混合层，特别是超音速混合层是否有效。同时还研究了大尺度结构是否会引发小激波，以及小激波对流场性质的影响等问题。这对建立超音速流的非线性稳定性理论是十分必要的基础研究。
由于在超音速条件下，实验比较困难，因此，本论文所采用的方法是直接数值模拟。
具体完成的工作可分三个部分。
第一部分,通过对小扰动演化的研究比较了三种计算方法的精度。
由于无法与实验进行直接比较，为确保数值模拟的可靠性，对三种精度不同，对激波的捕捉能力也不同的计算格式，即二阶精度的NND格式、三阶精度的弱迎风紧致格式和五阶精度的弱迎风紧致格式进行了检验。计算了小幅值T-S波的空间演化并与线性稳定性理论进行定量的比较。也研究了引入有限幅值扰动对混合的影响。结论是：对第一个问题，通过与线性稳定性理论分析解的定量比较可见，当T-S波幅值小于0.01时，五阶弱迎风紧致格式计算的结果与理论解相近，NND格式精度较差。当扰动幅值大于0.01以后，由于非线性的作用，计算结果与理论解偏离增大，说明超音速自由混合层当扰动幅值大于0.01以后其演化就不再满足线性稳定性理论了。对第二个问题，对引入扰动后所激发的大尺度结构而言，三种计算格式在定性上没有区别，在定量上差别也不大。
第二部分
1． 提出了两种新的研究方法确证了扰动会导致小激波的出现
是否能将不可压流的稳定性理论推广应用于可压缩流，关键是流场中是否会出现小激波。本论文研究了超音速混合层中扰动是否会引发小激波的问题，证实了的确会引发小激波。随之研究了小激波对扰动速度分布的影响。
为了令人信服地证实小激波的存在，提出了两种新的判断激波存在的方法。
为了研究激波对扰动的影响，准确判断激波位置是前提。通常确定激波的位置是从某一变量等值线上去找等值线密集之处，再在可能是激波的两边用激波条件去检验。但对弱激波这种方法不太可靠。似乎也可以看变量沿某一与激波相交之线是否有突变来确定激波位置。但对弱激波这一方法同样不敏锐，因为激波前后均不是均匀流场，通过激波的变化很弱时往往不易从变量沿该线的变化中分辩出来。
考虑到我们研究的问题中，小激波是由扰动引起的，其传播速度和扰动传播速度基本一致。而扰动传播速度≈1（无论从线性稳定性理论还是由我们的数值计算结果都是如此），故激波运动速度c≈1。而且激波的方向基本上与流向垂直，接近于正激波，因此在随激波一起运动的坐标上来观察流动就会看到混合层高速边流体沿流向而低速边流体沿反向穿过激波，即，高速层的波后在激波右面而低速层的波后在激波左面。本文针对所研究的具体情况提出了一个确定非定常激波位置的判断方法。
若某一时刻t速度场为u(t,x,y)，音速场为a(t,x,y)，则满足关系式：
, 
的点的集合包含着t时刻的激波。其中ε为可调参数。Δx，Δy的大小应根据计算所得激波厚度及方向选取。在本文中，由于激波走向基本上与流向垂直，所以Δy可取为0，而Δx则取为三个网格宽度。计算表明ε在0.005-0.01间变化时得到的激波点集合是一样的，本文ε取为0.01。（当ε取值很小时，如ε=0.001，则由于数值的误差将不是激波的点也包含进来，而当ε取值很大时，如ε=0.04，则将强度较弱的激波丢掉了。）u(t,x,y)及a(t,x,y)均通过数值计算而得。用上述方法确定的激波的确和密度等值线密集处一致。
为了更令人信服地验证是否是激波，在已经按上述方法确定激波位置（可以随时间变化）时，在激波波前指定一点，随该点运动並检验其在跨过激波时的熵的变化。由于流体质点的熵在激波前及激波后都几乎保持为常数，只有通过激波时才有突变，所以这种方法可以确凿无疑地证实激波的存在。
2． 分析了小激波的存在对流场的影响及小激波强度与入口处扰动幅值的关系。
近年来，有人试图将不可压缩流的流动稳定性非线性理论直接套用到超音速流中来。本论文分析了激波前后扰动速度剖面的变化，还研究了入口处扰动幅值与激波强度的关系。小激波强度在入口处扰动幅值小于0.05时随扰动幅值线性增加。小激波的存在会使激波前后扰动的形状有显著的差别，小激波强度越大则扰动速度分布形状改变越大。而在无激波区显然扰动速度分布的变化是连续和光滑的。从所得结果看，不能简单地将不可压缩流的流动稳定性非线性理论直接推广用于有小激波的超音速流。因而，研究小激波对扰动演化的作用是一个重要的课题。
第三部分
提出了在超音速混合层中激发大尺度结构的新方法
由于马赫数较高时，实验十分困难，因此本论文用直接数值模拟的方法对超音速混合层中大尺度结构的新的激发方法进行了研究。
对混合层通过外加激励以增强混合效果，在理论上和工程技术中都有重要意义。通常的方法是在混合层中引人不稳定波以控制大尺度涡的发展以控制混合。但此方法对超音速流作用很小，因为超音速混合层的不稳定波增长率太小。Wang & Fiedler的在圆管中的不可压混合层入口处低速部分加振荡的实验表明，在一定参数下引入振荡可极大地提高混合效率。对二维亚音速可压混合层的低速入口部分加入沿流向的振荡数值模拟结果证实了这一方法对亚音速可压混合层也能增强混合。但对于超音速流是否有同样的效果，是一个值得探讨的问题。
在实际混合装置中，扰动引入的实际操作是否容易实现、展向涡的发展是否快、其饱和后的涡尺度是否大，这是衡量某种扰动方式能否增强混合及是否实用的主要因素。
本论文比较两种方法，即在混合层入口的低速部分强迫其沿流向振荡，和在入口处引入T-S波，看其对超音速混合层中大尺度结构的激发及演化的影响。确认了新的激发大尺度结构的方法在一定范围内是有效的。系统研究了其幅值、频率以及混合层的速度比和对流马赫数等参数对混合的影响。
结果表明对于对流马赫数小于１的超音速混合流，在低速入口部分加入沿流向的振荡或在入口处引入T-S波均能增强混合，但前者比后者更有效。此外，还比较系统地研究了引入的振动的幅值、频率以及混合层的速度比和对流马赫数等参数对混合的影响。
对二维超音速混合层，在各种参数相同的情况下，与在入口处引入T-S波相比，在低速部分加入沿流向的振荡这种激励方式，尽管输入功率和产生的激波强度较大，但混合层中涡的发展快，而且引入振荡有实际可操作性。相比之下，要引入大幅值TS波，操作起来更不容易。
振动频率是影响混合层展向涡发展的重要参数。频率越低涡的尺度越大，但出现的地方则推后。当振动频率低于0.05时，开始在混合层中激发出小尺度涡，而且频率越低则小尺度涡的个数增多。小尺度涡的出现对混合有利。任何装置都只有有限长度，可根据涡的尺度及涡出现早晚这两个因素，选择一最佳频率。
对一定的速度比Ra，对流马赫数Ma越大混合越不好。当Ma大于1时，对所有的频率，都无法激发大尺度涡。而对一定的Ma，Ra越小混合越不好。
展向涡的出现随着振动幅值的增大而加快。但展向涡的饱和尺度并不随着振动幅值的增大而不断增大，说明其演化最终取决于混合层本身的内在性质。但这种性质是非线性的性质，而不能用线性稳定性理论来解释。因为即使按线性稳定性理论是稳定的，但引入的扰动有较大幅值时，仍能激发大尺度涡。
关键词：compressible
mixing layer enhancement of
mixing
shocklet
theory of hydrodynamic stability direct numerical simulation
可压缩剪切层 强化混合 小激波 流动稳定性理论 直接数值模拟
On the enhancement of the
mixing of a 2-D supersonic mixing layer
Abstract
The
enhancement of the mixing in supersonic flows is an important problem for the
future development of aeronautical and aerospace technology. For a 2-D mixing
layer, according to the linear stability theory, when the compressibility
effect increases, the instability of the mixing layer becomes less and less
obvious. Thus, it will be more difficult for the generation of large scale
vortices, making the mixing less and less efficient. It is difficult to enhance
the mixing effect by merely controlling the development of instable wave.
Therefore, new method is necessary to excite the large size structures for
supersonic mixing layer.
In
1993, Wang & Fiedler found a new method, which was different from the
traditional method, and was able to greatly enhance the mixing for
incompressible mixing layer in a tube. In this paper, the primary goal was to
see whether the same method is effective for compressible mixing layer,
especially for supersonic mixing layer. In addition, we also studied if
shocklets would be generated due to the generation of large-scale structures,
and how the shocklets would influence the flow characteristic. This study is
necessary for the development of a nonlinear stability theory for supersonic
flows.
Experiments
for supersonic flows with high Mach number are extremely difficult. Therefore,
studying this problem, we used the method of direct numerical simulation.
The
work in this paper consists of three parts.
The
first part: Three
numerical scheme were checked by comparing their results for the evolution of
small amplitude T-S waves with result obtained by linear stability theory.
Because it is very difficult to
compare the computational results with those obtained by experiments, in order
to check if the numerical scheme used was accurate enough and if the program
was correct, three numerical
schemes have been used, namely, the NND scheme, a 3rd order and a 5th
order, weakly upwind, compact scheme. The spatial evolution of a small
amplitude T-S wave, introduced at the inlet of the mixing layer, was calculated
by using these three schemes, and the results were compared with those obtained
by linear stability theory. The effect on the mixing by introducing disturbance
with finite amplitude at the inlet has also been investigated by using these
three different schemes. The conclusion was: for the first problem, the 5th
order compact schemes yield result comparable with the linear stability theory,
while the NND scheme was not satisfactory for this purpose. When the
disturbance amplitude exceeded 0.01, due the non-linear effect, numerical results
deviated from the theoretical results, imply that when the amplitude of
disturbance in a supersonic mixing layer is larger than 0.01, its evolution
would no longer follow the linear stability theory. For the second problem, the three schemes yielded
qualitatively the same result, and quantitatively, the difference is not big.
The second part
1. Two new methods were proposed to verify the existence of shocklets
induced by the disturbance in a 2-D supersonic mixing layer
The
problem of whether a simple extension of the nonlinear theory of hydrodynamic
stability, good for incompressible flows, to the case of supersonic flow is
appropriate depends on if shocklets would exist in the flow. In this paper, the existence of the shocklets resulted from the disturbance in a
2-D supersonic mixing layer is investigated. It is verified that shocklets do exist.
Also, the influence of shocklets on the velocity distribution of the disturbance was investigated.
In order to be reliable to confirm that shocklets really exist, two new methods for the determination of shocklet location is proposed in this paper.
In order to study the effect on the shocklets, one has to accurately determine their location. The locations of shocklets were usually determined by finding the densest parts of the iso-contour plot of a certain flow variable and then checked by if the parameters on both sides of the shocklets satisfy the shock relation. However, this method may not be reliable when the shocklet is weak. It seems feasible to locate the shocklets by watching the sharp change of a certain flow variable along a line intersecting with the shocklets. However, it also becomes unreliable for weak shocklets due to the fact that the flow field on both sides of a shocklet is already uneven, thus it is not easy to pinpoint the small jump of the flow variable among the all changing flow field.
For our cases, shocklets were induced by the disturbance, thus the propagation speed of the shocklet is very close to the propagation speed of the disturbance. Let c be the speed of shocklet, then we can see that in the high speed side the fluid particle would pass through the shocklet in the positive stream-wise direction, and in the low speed side the fluid particle would pass the shocklet in the opposite direction. That is, the shock front is in the right of the shock in the high speed side of the layer, while the shock front is to the left of the shocklet for the low speed side of the layer. If the flow velocity and sound speed are denoted by u(t,x,y) and a(t,x,y) respectively, then the shock can be determined as the set of the points satisfying the following relation:
In which, εis a certain parameter and its value will affect the numerical thickness of the shocklets. It was found that for our cases a value between 0.005 to 0.01 forεmight yield good results. Then,ε is fixed as 0.01. u(t,x,y) and a(t,x,y) were obtained from numerical computation. It is confirmed from the computation that if the Mach number is not too large, for a wide range of Ra, the most unstable T-S wave introduced yielded a c≈1 as predicted by stability analysis. The location of shocklets established by the method above is consistent to the densest part of the iso-density contours plot.
In order to make it even more reliable, one effective method to examine whether it is a shocklet or not is as follows: after the location of a shocklet has been determined by, say, the above method, which is changing with the time, one can follow a fluid particle, starting somewhere near and in front of the shocklet, to see the entropy change of that particle. The entropy of the particle remained almost unchanged when it did not meet the shocklet as in an adiabatic, isoentropic flow, but would unambiguously have an entropy jump when the particle passed through the shocklet. By this criterion, shocklets found by the previous method were proved to be correct.
2. The influences of shocklets on flow characteristics and the relationship between the strength of induced shocklets and the amplitude of disturbance the inlet has been studied.
In recent years, attempts have been made in applying the nonlinear theory of hydrodynamic stability good for incompressible flow to supersonic flow. In this paper, we studied the influence of shocklets on flow characteristics and the relationship between the strength of induced shocklets and the amplitude of disturbance at the inlet. It was found that the shocklet strength increased linearly with the disturbance amplitude when its value less than 0.05. The velocity profiles in front and behind a shocklet had appreciable change. The greater the shocklet strength is, the greater the change will be. While in the no shocklet zone, apparently, the distribution of the disturbance velocity would change continuouslly and smoothly. From our results, it is inappropriate for a simple extension of the non-linear theory of hydrodynamic stability, good for incompressible flow, to the case of supersonic flow. Therefore, to study how the shocklets would influence the evolution of disturbances is an important problem.
The third part: A new method for the excitation of large scale structures in supersonic mixing layer has been proposed
Experiments for supersonic flow with high Mach number were extremely difficult to conduct. Therefore, the new method for the excitation of large scale structures in supersonic mixing layer was studied by direct numerical simulations.
The enhancement of the mixing for a mixing layer by applying excitation is an important problem both in theory and technical application. In Wang & Fiedler’s experiment (1997), the inflow speed at the low speed side of an incompressible mixing layer, which was confined in a tube, was forced to have an undulation, and with appropriate parameters for the undulation, the mixing effect was greatly enhanced. The numerical simulation for a 2-D subsonic mixing layer, in which the in-flow speed at the low speed side was made to have periodic stream-wise undulations, showed that this method was also effective in enhancing the mixing. Nevertheless, whether the same method is effective for supersonic mixing layer remain to be a problem worth for studying.
For any practical equipment, the criteria for the feasibility of the proposed method should be the easiness of its implementation, the rate of growth of the excited vortex and the saturated size of the vortex.
In this paper, numerical simulations have been done for a 2-D supersonic mixing layer with two different types of excitation, namely, the introduction of a T-S wave at the inlet and the enforcement of the inflow speed at the low speed side to have periodic stream-wise undulations. The results showed that both methods were effective when the convective Mach number Ma was less than 1, but the method of periodic forcing of the stream-wise speed at the low speed side was more effective than the method of introducing a T-S wave was. Systematic computations have also been done for analyzing the effect of different parameters, such as the amplitude of the undulation, the frequency of undulation, the velocity ratio as well as the convective Mach number, on mixing.
For a 2-D supersonic mixing layer, if all other parameters keep the same, then compared with the method of introducing T-S waves, although the method of forcing the inflow speed on the low speed side to undulate would require more power input, but the induced vortices would grow much faster, and the method is practically easier to be realized. The introduction of large amplitude T-S waves would be more difficult from the practical point of view.
The frequency of the undulation plays an important role in determining the development of the span-wise vortices. The lower the frequency is, the larger the size of the vortices would be and the quicker the vortices would be generated. When the frequency is smaller than 0.05, small-scale vortices would be generated, and the lower the frequency is, the more the small-scale vortices would be. The generation of the small-scale vortexes is in favor of mixing. For practical equipment having a limited size, there must be an optimal frequency in regard with the size and quick generation of the vortices.
If Ub is fixed, the larger the value of Ma is, the worse the mixing would be. If Ma is greater than 1, then large-scale vortices can hardly be generated for all frequencies. If Ma is fixed, then the smaller the value of Re is, the worse the mixing will be.
The appearance of the span-wise vortices would be quicker as the amplitude of the undulation gets larger. But the saturated size of the span-wise vortices did not keep increasing as the amplitude of the undulation becomes larger. It implies that the evolution would eventually depend on the intrinsic property of the mixing layer itself. This property is of course non-linear in its nature, not explainable by the linear stability theory. For example, even the flow is stable according to the linear stability theory, large scale vortices can still be generated as long as the amplitude of the introduced T-S wave is large enough.
Key
word: compressible mixing
layer enhancement of
mixing shocklet theory of hydrodynamic stability direct numerical
simulation
