Q:

Centripetal Force Acceleration

In uniform circular motion, acceleration is $\frac{v^2}{r}$ and time which it acts $\rightarrow 0$.
So $\Delta v = 0$, but then why/how does direction change, when the acceleration should be producing no effect at all .

A:

In polar coordinates, the velocity being tangent to the circle, it is directed along the $\hat{e}_{\theta}$ vector. The centripetal force is directed along the $\hat{e}_r$ vector.
So 
$\frac{F}{m}\hat{e}_r = \frac{d\vec{v}}{dt} = \frac{d(|v|\hat{e}_\theta)}{dt} \underbrace{=}_{\text{Chain Rule}} \frac{d|v|}{dt}\hat{e}_{\theta} + |v|\frac{d\hat{e}_{\theta}}{dt}$
Now in polar coordinates, the direction vectors are not fixed, they are linked to the body so they have a time derivative. In this case, you can easily convince yourself that the time derivative of of $\hat{e}_{\theta}$ is something directed along $\hat{e}_r$. If you fail to see this, write out the polar vectors in cartesian coordinates, and derive them with respect to time assuming $\theta$ is a function of time.
The above equation can then be pictorially seen as
$\text{Force along}\hat{e}_r = \text{Change of norm along}\hat{e}_{\theta} + \text{Change of direction along} \hat{e}_r$
The force being always directed along $\hat{e}_r$ can only change the direction of the velocity vector. It should have a component along $e_{\theta}$ to change the norm of the velocity.
An intuitive picture to your question would be this : for a very short time, the force being orthogonal to the velocity, it will only shift it's direction. As soon as this change of direction occurs, the force is again orthogonal to the velocity, and so on. 

A:

Officially, I completely agree with the other answer given. I would like to offer this answer as a simplistic, intuitive answer to the question. No math involved.
I understand where your question comes from. In fact, depending on your current level of education, this question could indicate a high potential for future scientific success.
We all know that if you throw a ball, it falls to the ground. Gravity acts downwards and it make the velocity of the ball change to a more downwards direction. So why doesn't it do the same for things in orbit? (That's where gravity is the centripetal force). Why doesn't it change their velocities to more inwards directions?
The answer is quite simple, it does! Circular motion is a very special case. Imagine Superman. Say he wanted to show you how it was done and he threw the ball at around 8km/s. Naturally, you'd expect gravity to make the ball's velocity shift towards the ground, and you're right, it does. However, the ball is moving so fast the by the time gravity has shifted the velocity towards the ground, the direction of the ground has shifted by the same amount (due to the curvature of Earth), which means the ball still seems to be moving parallel to the surface, even though gravity has made the velocity shift downwards.
Ok, that's no problem. I'm sure you knew that already. So why doesn't the speed of the ball ever increase or decrease? That is simple too. Say the moment Superman throws it, the ball is moving horizontally starting from the top of Earth (say at $0^o$). The very next instant, you'd think there should be that same horizontal velocity plus the new, very small vertical velocity that gravity just added, right? No. The instant it is thrown, gravity acts completely downwards. However, as it travels to the next instant of time, the ball will experience a very small amount of force in the horizontal direction, against the direction of motion because there were non-zero angles along that path and gravity, acting towards the center of Earth, had a small component that was in the horizontal direction. The effect of this is to slow down the horizontal velocity and speed up the vertical velocity such that the ball is travelling at the same speed and still parallel to the surface.  As I said, circular motion is a special case. If the ball were thrown at any other velocity, the speed at point 2 would be slightly different and eventually it would hit the ground or fly off into space (or elliptically orbit).
So you see, centripetal force is accelerating the velocity towards the center, but as said previously, the tangential velocity of the object is just right so that the object is always in the right place at the right time; it can maintain the same speed tangentially without accumulating radial velocity because the centripetal force and tangential velocity are in perfect balance.
But, you say, when I swing a ball on a rope, I can make it go as fast as I want and it keeps the same radius. How can more than 1 speed be the perfect speed? Well, in that case, the centripetal force is the tension in the rope. The faster you twirl it, the more tension there will be. The net effect is that the force is constantly changed so that the velocity is perfect.
Hopefully, that answered the question.

A:

By Newton's law ${\bf F}=md{\bf v}/{dt}$:
$${\bf F}\cdot {\bf v}=m\frac{d{\bf v}}{dt}\cdot {\bf v}=\frac{d}{dt}\left(\frac{1}{2}mv^2\right)$$
The above says that "the applied force on an object times (dotted) with its velocity is equal to the time rate of change in its kinetic energy." Notice that if the applied force is perpendicular to the velocity, that is if $${\bf F}\cdot {\bf v}=0$$
then the kinetic energy is constant. This is the case in uniform circular motion, and hence the object doesn't speed up.

