RadialLineRenderer.cs

using UnityEngine;
using System.Collections;

public class RadialLineRenderer : MonoBehaviour
{
    LineRenderer lr;
    [Range(-1f,1f)]public float wave = 0.5f,spin = 0.5f;
    public float radius = 3f;
    public int verts = 500;
    [Range(2,8)]public int armCount = 5;
    [Range(0.5f,16f)]public float armPower = 0.5f;

    void Start ()
    {
        if (lr == null)
            lr = GetComponent(typeof(LineRenderer)) as LineRenderer;
        if (lr == null)
            lr = gameObject.AddComponent(typeof(LineRenderer)) as LineRenderer;
    }

    void Update ()
    {
        if (lr == null)
            return;

        lr.SetVertexCount(verts);
        for ( int i = 0 ; i < verts ; i++)
        {
            Vector3 p = transform.position;
            float a = ((float)i / ((float)verts-1f)) * 2f * Mathf.PI;
            float mod = Mathf.Pow(0.5f+0.5f*Mathf.Sin( a * armCount + Time.time * spin * 2f),armPower) * radius * wave;
            float r = mod + radius;
            p.x += r * Mathf.Cos ( a );
            p.y += r * Mathf.Sin ( a );
            lr.SetPosition(i , p);
        }
    }
}